Class 9th Mathematics Term 3 Tamilnadu Board Solution
Exercise 3.1- The radius of a circle is 15 cm and the length of one of its chord is 18 cm.…
- The radius of a circles 17 cm and the length of one of its chord is 16 cm. Find…
- A chord of length 20 cm is drawn at a distance of 24 cm from the centre of a…
- A chord is 8 cm away from the centre of a circle of radius 17 cm. Find the…
- Find the length of a chord which is at a distance of 15 cm from the centre of a…
- In the figure at right, AB and CD are two parallel chords of a circle with…
- AB and CD are two parallel chords of a circle which are on either sides of the…
- In the figure at right, AB and CD are two parallel chords of a circle with…
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- In the figure at right, AB and CD are straight lines through the centre O of a…
- In the figure at left, PQ is a diameter of a circle with centre O. If ∠PQR =…
- In the figure at right, ABCD is a cyclic quadrilateral whose diagonals…
- In the figure at left ,ABCD is a cyclic quadrilateral in which AB || DC. If…
- In the figure at right, ABCD is a cyclic quadrilateral in which ∠BCD = 100°and…
- In the figure at left, O is the centre of the circle, ∠AOC = 100° and…
Exercise 3.2- O is the centre of the circle. AB is the chord and D is mid-point of AB. If the…
- ABCD is a cyclic quadrilateral. Given that ∠ADB + ∠DAB = 120°and ∠ABC + ∠BDA =…
- In the given figure, AB is one of the diameters of the circle and OC is…
- In the given figure, AB is a diameter of the circle and points C and D are on…
- Angle in a semi circle isA. obtuse angle B. right angle C. an acute angle D.…
- Angle in a minor segment isA. an acute angle B. an obtuse angle C. a right angle…
- In a cyclic quadrilateral ABCD, ∠A = 5x, ∠C = 4x the value of x isA. 12° B. 20°…
- Angle in a major segment isA. an acute angle B. an obtuse angle C. a right…
- If one angle of a cyclic quadrilateral is 70°, then the angle opposite to it…
- The radius of a circle is 15 cm and the length of one of its chord is 18 cm.…
- The radius of a circles 17 cm and the length of one of its chord is 16 cm. Find…
- A chord of length 20 cm is drawn at a distance of 24 cm from the centre of a…
- A chord is 8 cm away from the centre of a circle of radius 17 cm. Find the…
- Find the length of a chord which is at a distance of 15 cm from the centre of a…
- In the figure at right, AB and CD are two parallel chords of a circle with…
- AB and CD are two parallel chords of a circle which are on either sides of the…
- In the figure at right, AB and CD are two parallel chords of a circle with…
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- Find the value of x in the following figures.
- In the figure at right, AB and CD are straight lines through the centre O of a…
- In the figure at left, PQ is a diameter of a circle with centre O. If ∠PQR =…
- In the figure at right, ABCD is a cyclic quadrilateral whose diagonals…
- In the figure at left ,ABCD is a cyclic quadrilateral in which AB || DC. If…
- In the figure at right, ABCD is a cyclic quadrilateral in which ∠BCD = 100°and…
- In the figure at left, O is the centre of the circle, ∠AOC = 100° and…
- O is the centre of the circle. AB is the chord and D is mid-point of AB. If the…
- ABCD is a cyclic quadrilateral. Given that ∠ADB + ∠DAB = 120°and ∠ABC + ∠BDA =…
- In the given figure, AB is one of the diameters of the circle and OC is…
- In the given figure, AB is a diameter of the circle and points C and D are on…
- Angle in a semi circle isA. obtuse angle B. right angle C. an acute angle D.…
- Angle in a minor segment isA. an acute angle B. an obtuse angle C. a right angle…
- In a cyclic quadrilateral ABCD, ∠A = 5x, ∠C = 4x the value of x isA. 12° B. 20°…
- Angle in a major segment isA. an acute angle B. an obtuse angle C. a right…
- If one angle of a cyclic quadrilateral is 70°, then the angle opposite to it…
Exercise 3.1
Question 1.The radius of a circle is 15 cm and the length of one of its chord is 18 cm. Find the distance of the chord from the centre.
Answer:![](data:image/jpeg;base64,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The figure is attached above
A is the centre of the circle.
AB is the radius = 15 cm
CD is the chord = 18 cm.
We need to find the distance of the chord from the centre i.e. AE
In this circle, we draw the perpendicular
We know that perpendicular drawn from the centre to the chord, will bisect the chord, such that CE = ED =
= 9 cm
Now,
In ΔAEC,
Applying Pythagoras theorem,
AC2 = AE2 + EC2
⇒AE2 = AC2 – EC2
⇒ AE2 = (15cm)2–(9cm)2
⇒ AE2 = 225 – 81 = 144
⇒ AE = √144
⇒ AE = 12 cm
∴The chord is 12cm away from the center of the circle.
Question 2.The radius of a circles 17 cm and the length of one of its chord is 16 cm. Find the distance of the chord from the centre.
Answer:In the given figure:
![](data:image/png;base64,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)
Radius AB = 17cm
Chord CD = 17cm
Now we draw a perpendicular bisector of CD from A which is AB
Hence CE = DE = 8cm
We have to find the distance of chord from centre i.e. AE
In the triangle ACE
AC2 = AE2 + EC2
Therefore AE2 = 172– 82
AE2 = 225
AE = 15cm
Hence the distance of chord from the centre is 15cm
Question 3.A chord of length 20 cm is drawn at a distance of 24 cm from the centre of a circle. Find the radius of the circle.
Answer:The figure is given below:
![](data:image/png;base64,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)
CD = 20cm = length of chord
AE = 24cm = distance of chord from centre
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,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)
From here we get,
r2 = 100 + 576 = 676
r = 26cm
Hence radius is 26cm.
Question 4.A chord is 8 cm away from the centre of a circle of radius 17 cm. Find the length of the chord.
Answer:The figure is given below:
![](data:image/png;base64,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)
AE = 8cm
AB = AC = 17cm
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,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)
d = 8cm
r = 17cm
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence the length of the chord is 30cm.
Question 5.Find the length of a chord which is at a distance of 15 cm from the centre of a circle of radius 25 cm.
Answer:The figure is given below:
![](data:image/png;base64,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)
AE = 15cm
AB = AC = 25cm
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAvCAMAAAD3nYnBAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmY6Oma2OpDbZgAAZgA6ZjoAZjo6ZrbbZrb/kDoAkGYAkGZmkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bpRHQVgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACHElEQVRYR+1Ya1eDMAxNp67TDXWiG9M6H3T0//9DSyEtr3paYNkH5dN2DjQ3N4+bFOD/mZ+BrwfG1vMfG3Nicb+D42If88lZ3pWry2MQm7O45jtUsD7vR+p0yHsYBDUE6GHINYT8iSgY6uVVW6oxSHS+SJh+qDAcTOYhD9U/4kcsjUHEoFIq352f2ANsPuRXb8QsQFbR4HJSpdTRsK3Q1QU5EeLmu8MDFAktEc5eoz9kiIsmLxoRcL2g3zLHgAnWfoFFkDMnGJKPCEbXZLD2q3SIdpXWtRLh/qDJIO0vkkFrcQmhKazaWs9kkPZ7WM/i2lReY+iaDNN+/LpDuvDMcJ4+LrnhoWsyUPvnxFCbLCPzzrUTodrvcdgDDQZ4kHr6vit5sCYlv91+rPbB2j8Zg+RbUM86H5xJb2Vn5UBinlaoJ2MwFdSmrc6O4LpGDBYhqzR0YH4a9KJq9W2rYzF0QAfnQxSGM8UiCoMnOlNrs+rq7VN8JPoSZCoGEIsdnB6b6SM5Y1GSF9mr+/1BHdhi/cmn7ABWs9SH7jXXuOHGaVZwFQ6/aLU7Y1s4JbVUjdHuCUBwhjHTtUANHjPDjAfRGtswQ+eZ5YJBtWZoHBtI06Hk3w1z2GSpZ/vG/GX3K/Idx+56thjodz1LRLbEhStumAxOvd9erHZ/kxe5brKX2P0BylsPM5arTP+6yB0IlHdBtbZvwN4FzcLx3z3kB1ajKQ9strxWAAAAAElFTkSuQmCC)
Here d = 15cm and r = 25cm
![](data:image/png;base64,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)
![](data:image/png;base64,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)
l = 40cm
Hence the length of the chord is 40cm.
Question 6.In the figure at right, AB and CD are two parallel chords of a circle with centre O and radius 5 cm such that AB = 6 cm and CD = 8 cm. If OP ⊥ AB and CD ⊥ OQ determine the length of PQ.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALsAAACuCAYAAABui3xCAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAACYPSURBVHja7Z0HeBTVFsejPt9DESuKoCKiICKCDQFFpClKE5SqFEEUsYHYEGy0EJKQnhBSKAmhhRISSCCUQCAQQk+ANCCJkACZsrN9Z3dm/+/OREEkQkjf3fv/vvstCZstd35z5pw7557jBioqF5EbnQIqCjsVFYWdiorCTvUPyRcE2LLOw3aq5Mo4WQxbzkXIRpFOEIW9PssO2GTInAnWI3/AnJgF07J0GOZvg+6XBGhHx0LTNpKMJergnwoH/3go+BaLrh5PkPH0X89bDOH1aGi/3QDdz5tgDNwF06pDEHflQTrNwq63krNGLntvKgp7jWAtS7CbrJDPaSHuPgPT0gPQ/bARmq6LwTcPBfdoCLimweAeDgL3UCAZAeBbh0AYsgzCiGj1UfvFOhhmJcMwZxsMs/8cc7dB/9sWCCOXQxgWBWH0cmi6LwbX2B/cg+R1mpDRlLzmIyHgH1uonizCqFXQz06CcdkBcpIVw66xwG610ROAwl5ZyQRuC3E99BC35EE/Zys0fZYSmBUAg8rAbkYAbBsK7RgCn3syTIvTIW7Lg+1oMeSLRthZE+xm8cqwKlZZumbYJenKcyxk6C3q38uFAqxphTDHZcEYmgbd9E0QPlwBvtWiv51UZDQPhnbiGhhD9sJ64Bxk3lT2XlQU9n+XHbLOBCmXhXFROoQxK8G1Diyzss2CiKsRDu3YNTAE7YKYnAO5SENOCGJRbRJq3qralcsL7OS97AYrbIfOqe6Nfi65KvSPJleXheqJyN7nC+7lUOh+2gRLYi7kEh0Bn8YDFPa/MDKaIZ0qhSk0g7gTK8E2XkAA91MtqPD2KhjC90BMOw1ZMNXPLyDbIZ1lScyQCf30JGheWUbcnRCwDb3AtQ8m4CfCsjEX0jlycso2Srkrwi6d52BZewq6z+LAPuQN9p4F4NsQ6z16PbGaGZAKGQf9YhLEg2dh8NoO4U3i9jweDOY2D/DE4ht+30ncrPOQtQYKu7N/QWIDYTteDKNfOvjXwsDcOY9Y8BBoh6+HMSYDtjMXnev72mwQD5wm4O9UV3nYh/zAPOgF7ahYWJZlQrqoobA7HeSyFeJuYu2m7wDbzBeMmzs0ry2DYX4KxGMF6kng7JJZLcxbs6D7IIGc4MTa3zoXwjtRMAWRk7zAuU5yl4TdDmLZduVDP2kzmEaeYP47D9rBsTCFH4RUyrvcAf5L1mOFMPruAf9iOErdZoHvEgaTT5pLQe9UsFsPFUH38WawDywAc4s7cVXWwrIhC3ZLPQ0062KO8ktg9E8j0C9Sodd0iYApYD8kXqCwO4JseSUw/LwD/FPkUu0290/IT0I20KDsX6HPJdD77QP3wkKU3k7cm7eWw7ziKOyS8y5bOjTsst4IU9QBaJ6PIFZqNoR3Y2BJyIas1VOaK2woLsAYuB9sc38w/5sP7bC1EPecprDXJ4mHz0A7YA2YOz3APR8Cc9hhSIzzX4prDPrjJdB/l0jcPw8CfiAMnjshXXSuGMfhYJd5HfQztoN7Ighs0wXQT0+G7cwlSms1yC5aICbmQjM4GkxDT2jaLyZXyiwKe11IPFIAYcBKMLfPg6bPMlj25tOckBqQzOth9EkF29hPvbOs/3UL+Z3jxz8OAbuSMGUK30+seSCx5r4wzk8hLouOUlmTcy5LsB4sgjBkOdg7vIhxWQ7xQAGFvSZlO8dCO2YD2Lvmg38lDObkU+rtcarakawzwuCeQtyaBcTYBMO0/ABkq2Pm29Rj2GU1GUvTLRJsIy81n0UqYil9dXIoZFi2ZqvGhr3fG7ppmyGXaCjs1SElv8O0NANsqwBwjwfAFJGm5ppT1a2UPCLtmFgwDeZD6L8c1pMlFPYqGRGDGUb33WBuna/e2jYnn6CU1afjYzLBMGsn2Pu8wD8fDsuefDjKjql6Bbss6KH7cjMYtzkQ3iSW48Q5Sle9lB3mmKNgH/UD29wP5o2ZDpFYV29gl/4ggejHGwjo86AbEUcCUwfNLXchWXbkgHs+VL3fYQzfp7qfFPYbgV7EqD6gktein5IEu9lMSXIQWY8Wge8YBua/7jAG7KnXO6PqHHapsBTCgBg1FVf/7RbYDRR0R5N4pBCad6LA3ONJgE9V1+gp7P+QrfBSGej3esEYuJdadAeW7Ty5Og9bpe4hMPgT4O0yhf3y5BRchDBoBZj7Cejh+5V9/pQYB5dUzEEYsgrsPYrxSkV9W6WpE9jLrMBqlLrNgf7XbfXSClBVEvgSnlytl4O9zxuGIMWHrz/HttZhlwwG6D5LUJcX9V+TYLS+lqugqrRsOSXgX1kE9kEfmOOPuybsdosF+h+2ofQ/JBidnEiCUQq6s8p6WFmlCQXXMkitielisNthCt6P0tvdoR20BhJPsxadXZbteWAbLQDfPhzWk+ddB3ZL0imwD3hD88Yy2PJdr4yDywK/5riaIiy8sRxySd3ufKoV2G1Z58A9FwyuTQjE9DOUABeSXbJBPzcFzK3u0I3bCLtYdxu6axx2mRGgfX8lmLu8YF53jB59F5SS86T9dD2Y/82DMSQNdnvd3HSqWdglGYZp29VqXIYZ24nXTuuJu6qU3Ce++2K1OptlZ67zwW7ekEVA94B24GpIHA1IXV1iaoG6+UPTNUpdj3ca2G25F4ifHgS+wyJYT/xBjzQV1BW5wHSU3ja7LOGvlv33GoFd2SCtHbUWrJsnzKsP0WNMdVmSoIP2o3Vg7p0Py8ba3ZhTI7CbVx4G+18vaEesU9uoUFH9XdbcYnAtF6o7naRCznFhl4p5cM8rXyQUUgEtXkRVvkwhxJ25da5a5Kq22uJUK+x2ux36zzcR98UL5hjqvlD9u2SdAdqRsSRg9YKYnO94sFtSc9WGVsLQVarfTkV1PSk3GLnGARD6RcOuq/k8qWqD3W62QugVA/YeH1iPFNEjSXVjZmQJ+hlbwDT0gGnFkRrf01BtsJtiSVDawAe6KZvVPp9UVBWRdJGDpnUECViDSbBas2vv1QK7fFGAptNScK0CSFBKq3ZR3ZwMgXvA/M8LRs9dNbrZo1pgN4WnqZltxgV7QVuMU92sZIMJQo9o8E8FwVZQc0uRVYZdvsRD8/JS8F3DyCWJNgOgqpzMqw+Du9cPBo+dqCmDWWXYTYvS1ML1pkUH6RGjqrRkowWaHlHgngqBVFQz1r1KsEsMseodl4HrFFoniT1UziXz2iPg7veHwX1H/YPdFH0IzJ3zYZhLAgsb7YBBVTXZ9cS6d10Mvg0xnheq3yWuNOyy2Qzh7ViwT/jDdpqmBVBVj4zELWbv9IFxSXr9gd2SdFKtDaKfvkPdekVFVR2SSrTgWgcT/30Z7MbqvQtfKdgVl0X/eRLYRh4QU/LrxSRROYnsMvQzt4B7IBDi3urd0VQp2K3Z58A9GQTtu6sh62iDXarqlfXQebBNfaD7JL7uYTcFHUTpLbNgijxMjwxVtcsuWiEMXQ6++SLYiqqvTv9Nwy5rdBAGrFHLYljzi+mRoaoB2u0wRx0Fc4c3jBFpdQe7NeMsmLs9oJuwhTbcpaox2U6zYJv7QksMK6qp3vtNw27y3A+m0XxY4k/SI0JVY7KbLdBOWA++5SJY86qnK99NwS6zAoS+xJfqtAS2EtrziKpGcYc5OhPM/zxhithX+7Bbj50D09QLunFJ9aruNpVzSsorBdvCB9p311VLFbGbgt0ccgTM7e5qmgAVVU1L1huhHRULvl0krIVVL4ZbYdhlixm6iQngWy2E9RQtekRVO1KWudm7vWGOq3qd0ArDrtTq4zoEQ9t3ba2VPqCisqb9AabBPBi+2157sFt3FID5jzuMv6XQI0BVa7IVc+BfDYPw5mrIxqpVIKgY7HY7TL7pYBv7wrw1kx4BqlqT3UTc5wnx4J4OhXiyau5zhWBX+pPqvoiHpm04rHn0rilV7crkf0itDWneXDVDWyHYlYYCfPcwaF9bCbuRNuatDllEK/buTUNIYBCCAgIQHRWF7JxcOjHlSEw5A+a+eTDO21vzsEvZLJgHvKCflEhnvhp07Fgmvp36LUaPHosvPvscU6dMwYABAzGg/0Akb02mE/QP2c5eBN8+BLoPNqkpwDUKu3UTObMaeMIYtc9hJqi+Km3ffrzTpy/ef28Idu9Ovfz7nNx89O7RG++89TYKi+jS7t8l8Vpo310FoVs0pNLK73W+IexK92mj/z6wTQJg2VI/+lk6qjIzT+JtAvOgQYPxx7lrWyUG+gfiroZ3IXZNLJ2svzOobBaamAi+dTCsJypfWvHGsIsidN+TN2oVDls2DU4rK73eiFEfjMLTT7XGvv3l76+MDA+Hm5sbIsgj1dUyuR8A+6AXLOmV3xl3Q9hlnRHC8BgIL0TBrqcdqSurrcnb0a5NW3zx+Zew2q7ds2u1Sfjt19/x8INNkJCwiU7YP2RZfwqlt86GOajyG4ZuDDuvB987HNpXV6rr7VSV08yZs1WrvnlzUrn/X1hUhIEkSO3RrTuyTtD06X9KTD8N5h4PmH7ZX3OwS2c1YFv6QTdpI53xKmjI4Pfw6MNNkZOTU+7/x6xYiYceaIxpP0yDlW6KuUa2vGLw7RfCMCEJdqly83Nj2LNYsHd5w+C1jc54FTSwX3880uRhFBdfG/fk5eXjrTf7oGf3XjieSRcByuXwXCk0r0ZC98FG2K2WmoFd3HZWbTBgDN9DZ7wKmu/hiaeeaInY2HVX/f7MmQJ8MmEiOrRrj7i4eBednQrArtFC6L8Kmt7LIV3S1AzslpXHwT0cCEvCcTrjVZCydj6AWPeePXtj/fo4nMjKUh+HDx+J7q+/gbWxa+kkXUd2mwjdhxtV665Y+ZqBfQWBvQmBPf6Yw01QfdOpUzmYMeNnDOjbD33f6oORw0dgWfRyFBTStjw3hJ346bqxm8B3VNpJVm4jx/VhV24oBe0D90gwxJ3ZdMap6lSGb3eBfcwH1qOF1Q+73WaD/iflhlIYpDOlDjlBVM4j08wDYB70hJhxpoZgn5EE/ikCez6t1Ft5yZBJUCWVauhUVAX239PBPERhr7+Yc3qYFuyDdshKCMNWw7Ijh96co7A76QFKOAr27gVg3OaS4Q7uuYWQimnvKQq7E8oQmgq2kZfa4r5sLCABFk2oo7A7oSwpp8A+5E+sugcB3RuavtGQWVrmm8LuhLKbLLCsygTXNEi17NYDdE2dwu7kEl6KJtbdE/aLNE2awu7ssHeMUWGXi3R0Muoj7Gq9mIXp4JoFQdxGc6yrBPvLyynsVZIE/efJYFv4wXq8cq5gxXJjlESwTTQRjMJed7JLInSj4qHpFAGpqHJexo1hV4IrZbP1xqN0xinsdQe71VLzWY/izkKwd/vAGJpKZ5zCXmeSeKURxkpo+sRAYip3U+7GO5VOsGAbecMwnxbvobDXIezKTqUuf+5UslWuivSNYS8UwLb2h+7jODrjFPY6kzX3PLi2wTB8urXSHdUrVl2gTwSEV2KI30TbtlPY60ZiWj6YBzxgmptR6deoWN2YD4mv1H4JZEZLZ53CXieyrDoJ5h53mFdUflXwxrUeldJjv2wD/2QYbFnn6KxT2OtAdph+SQP7kDcsGadrEHalRV/YITWZyZxI19op7HWAumiB7qN48M+EwFqFhgQVq+K7pQDM7fNhjNzrMBNEYXceKTu8NG9HQ9szBjJX+b0AFavPfoYH+4Qv9ONoDUIKe+3Lml8Mro0/9GOU0oGV3+VVsc4bgh6awVEQOsfATgJWKgp7bUqMy0Pp7bNh9K1af4CK9VRSfKapiSRIXQjrsUKHmCBHhV2WZdjpHtWrZHLfD+Z+Epzuqlo5lwq2hiRBavhRsPd5wxx/lM5+NcMuaHWIi9+ESZMmoU/vN8noja++/AoHDx52wZn6x8lvNEH70Qbwz4Wr7kwtwA7YjhSDedQLhh+21/sJciTYs3NyMPT9oejftz9WrFiFXPLz/n3pGD1qLHp070mAP+TS82ZTmk2/tBDad2Jht1iq9FoV73B9SQDfazGE7isha+keypuG/ZVrN2/k5uWhV89eGDRwELKzc69yX1JT09DxpY747rsfodcbXHberLuKwNzuDsPPKVV+rQrDLit++5eJ4FoEEb+9gNL7l0QJUh5/w6FptxTMLV6XYWcYBmPHfITOr3RC2r5r286cO1+Cd/q8gz5vvY0LFy662qz+CZ0dRm+ln5efunG91mBXZIk8BuY/c2EKO0Ah/1NqVugtC9TN1IzbvH8dLLHq7H/9gPNlVjo6OgYtHntc7chRnpTueW/27K22i7x40TW3RMoaPTQDo8C/sBS2YrZ2YbfmlIB70h+6oUqapURJR1mzBsZtPri7A6DttvpfB9fIH+ztPsAFEwTRgMGD3kOnl1/B8eMnyn3dHTt348UOL2Dy5G9c1o2xnbwIpqk3dB9WT936m4JdNpmhHRVHzrRw2M656KX1n7CfYNVqX9o+16+vrvjsinVHiQlZ+dlo3uwRvD/4PcjytcuMym/mzJ6Lpk2aqlcASZaddPauI2X/c+RRMHd4wbS8ejwJt5v9A7N/BvkA7jDH0HrtV8He+/q9S5XVGI64OjhnxM59e9QuHJOnTC33uUoDsZeff1H12S9eclEXhhhWYdTqspakhdUzBzcNu/XEebCP+kI3LA6y2VLvJqk+w6749Sgx48SZHDzZ4gl8+MGoa57HMCzGjB6DZ9s8ix07Ulx2Xm3ZDAlMvaEdurbaCsG6VeaPLKMToXsiGOIp2nb8ZmBXlh5xwQyzbMOwoSPwTKunkZS09fJzjhw9jrFjxuC5du2xZs1a13RfFMkSjAv3g2nkAzm++hqqXRd2i0XE5s2J+PXnX/D1F19ixvQZOJqZiZMxe5DaYCqkQNe+4VEZ2O1/6FWf/NDhI+j7dj81SJ369RT1jumIER9i4qeTsHVLMlw5YcCut5AYJxyaVouxKyEJ30yZjNm/z8T0H6dh6uQpmPL115gzZy7iEzbDRNydKsO+O3Uvxo8dh6HEAinLY6EhC+HvH4Avvvwaw78cgzUtvofUd22VUi5dEfa/31RKTz+A11/rinFjP0JkRASBfCs0GlrO2pKcB67hAui/TcKJ3GzVEDS4/X8Y+t77CF8UhrDQUHz//TS80rETORGm4vz5ksrDvoVMerfXu2HIkKHq7Wr5bz5T1LJoDPlkNFIHBMF4rw8sW3Ip7Ddp2RWdPVsAX19/rFixGmbLtbvlL14sRfqBg9DqXCtL0i5J0H65DlyTIFgPnS3jMWkLWrZoiUTy+JcMBiMWhYajXdtnMW3adJgrED9eA/v+9Az06NYD/d7ph4LCazMcs3NysTh2DYpW74H20SDoJm2G3YUDVSmbK4P9zRvA3vGvANWI7KIz+Ojj8Wj5ZEu82bs3vp7yNZZERyPndNmWs9MFhfAnV9L9hw5ClF1rk7t0lgf3WCCEYSsB0QbRasXM32bitS6v4eixzKueqxjhYUOG4amWT+LgoRsnzV0Fu80mEZ/xMzzW9BFySS2/o7XZYoFWr4dEHnXvbQD7sDesx86rt3ZdcUiZTBnsvWKv+7y/lh6tp3mcPJ2LrOOZ2L1zF6b98BPe7TcQz7Zqg25diEszeiz8/YJx+NBRiIoRscN15lOSYfBOAXu3H0zrj6q8MSyHkSSWUdpoaoRrr3KzZs1Gy8efwJIlUVd5IDeE/cSJbHTt8ire6v0WeF5zwzPFnHQS7KP+4JsFg28RAf6xMPCPuth4aKF6B5W91ee6z1NSBZSuG9zDC1HaLATm1tEwtYpCyYOByGvgjgNu32Gf2zdIJ4+5DeZCaB4B4fEI15lHwo6mZQTYRj7QdFwMu7HMtTt58pTqm88kAWp58vCYj6efaoWoqJibs+xxGxPw/HPtMe3H6bBUIJ3SbjZB238NSt1mgb3NhwQVAeDudLFxVwD4xiHg7g++7vOUk0GF/Q5iHBoGlv2ezJf27mDo7wuF6YFwdRjvD4PunhDyHFebx0D1DjNzizdMi9KuMLkhDs+0fgYb4jaWy+D4j8ajxWPNsWt36s3BvjxmJdo90xbz5nnCWsGCSOYVB8E29Ibuq82QeQPsWpGOcobwQlkzAiWgpfNx7ZDOaaB5gVzJ2i+CdLGsPpFFJP7677OJt9GV+OtZ5caPr3bqjP79BhBPhL852JO2JKP9s+3w5RdfEdit1zxZJG9eXFysbh27HCTwOgg9YsA+4QfbWcZlA9Ub6XIzghLXzU2/nkzh6WDv9YMxRLHqZb63RqPB8GEjMWLo8GuS4URRxC+//Eqs/tNYujTqKiYrBLsSDIwYNoJEty2RnLwNknTlBc6fL8bq1WuwaFG4erPpKuu+MgNsA2Ldf0wiH9NF7/rdCHa64fpfJZVw4JsvAvd0CPn3lfsMyn2IDs91gJ+v/xXWSNCen38aHvM88NILL2HOXHcYjRVr3XPN0uPBQ0cw+N3B6Nmjl5p5t37dOsSuWYvffpsNf3/iV7HXXi6UepBCnxVgH/eFLYu2PaSw35z0c3aAvduL+Or7YbfLBGgz8dU3ol/f/nikycOY/tMM8vMGbFi/nnD4O94fMhz93+mH8PBI2GwVX5ot96ZSdk4eAgODMXHCJ/ho9Gh8MelzJCZuUZcm/02WXTlg7/SFduQaWgCVwl5h2ZTqvM1CoOm1GBJbNjeK+/LtN1Px2SefYvJXX+GzTz/FhHHjMX7sWPX34RGRKCi4+SoXbtX1oe1WCdqh6wjwC2BJce27qtUFe8LGeCwMDkZMdDRCgoIR4OeHoIAA1eopLqejq2yrZzzYxl6wJCg9u2o2I8itOl/MeuIcuNbB0PRYCpmjm7KrCnvU0qV47tl2aHTHnfj4o3H46ccfibWbiJdf6ojvv/sR54tLHHpOzLHHwTTwhO7TONhFa42/n1t1v6DeK0VdLzXMT6GEVxF2RXPd56HzK53B/c2S+yzwVdeWIyKWOOx8SKwATbcoNc6zHj5fK+9Z7bDLjA58j8XgWgTClknz3a+B/Y+Kw64st40dPRaffvoZdH9LCNuVshvNH3kEs2bNcdDZkGGcvQvMHfNgDE4jQWnt7Gd2q4kXFXfnqzeahLdXwm4wg+pvWY9cxecj4+ARdO7YCV5eC67ayKH83LZ1G6xavdYh50JMzQN3ry+EN5dD5muvwYVbzbysHYaft6s5IwbvXZR0BfbOyk0lDwhdV6npwNeMN9eqy7eWmCv1UVasXI1n2zyD2Nh16s96g0FNC1aqDkz95lu1bJ6jSSpmiREk7suD3rDsqN2FDLca+1KlWmh6Enfm8UCI+/Io7F1W/FlDxl3NkixvXHL7Gcbpey7/zfSfpuOlF1/GpElfYviQIejb52307dsfnp4LwLKsw82BXeni8hMxgre5w+CTquauOwXsisR9Z8E+Ri5Xr0dDOu/4S2VVgr3jcvVKZ91eBOkke80wzUlXYTfN2q8+XyNoMfjdQRg7dhyysk7g2LFjOHb0KAoLi9S0DUeUZXUmmAYe0A5YA1lX+6t1bjX78nYYfdPA3OoO3SfxxH83wVV12WdnyvfZxZhsAvuMy7AfyDis7k+dP9/LKb6/NfM8uKcDwT8XCtvRojr5DG41/QZKIwPdhA0k8p4PY8gewEW3Et9o6dGy+AQuuk2HbV7ZJvaw8Ai0IUHontQ9Dv/dpWIOmr5RZSXP4+quL5dbbbyJrYiB5s2lYJv7w7Kl5u+UOSrsrNuvuPh9MpKP7kXXLq+hw3Ptkbg5Uc0VcVQpZaZ13ySWVeKdsaNOP4tbbb2ReLiw7DL2dAjE9LMU9nJgN5IAdkPXWXh/3EgM6NsPgwYMxKD+/VFQUOCYoNskGBfsV4Ny3aR4yMa6PWndavPNLLGZYO/1Bt8lAtZc16oVWRHYFZ/dPDvdab6zaelBMLd5QugVTa7upXX+edxqfQJCMsglbY56w0lyoY7Zrga7ZXsO2Ed8wb8UXm+aRdc67LJohnneHhga+MIwcgMB3jVSXisK+1+rMQ4NekouuDbEZW1LXNZ9p+vN53Krizc1GXTYOyEM2W7TYRwZB5l1/gxJofOKsqVHQ/lr5GJsHoH9J5gXOHZJQQV0tnUguGa+wJ6CevXZ6gT2tRvi8MWvP+DI0IXQu3lBO2gtjD+nOfXgm4YS2L1g+Hpnuf+vGxBH/n8OtG+sdtzvOS0VHLHmfDM/aKPSsXFTPObMmg1vT0/M+v13/PzTdLVmY0L8ZpjqoLBWrcOek5OL7q+/gdHjPkbJyXyIX21GKTnISjmOUrfZxLr9ol7OnW0o3+2vlIDy/38WWDdfdS4c77v9ftUxtCw5BJtsw7adO/D+4PfR7MEmmPjJRPh4e6vluNsr+0r9AqCt5dyeWoW9lGEwdep3eOCeezFl8jcQrGaA00H7yXo1SYp7OBAm/wyIq3JhWX7KpYb+oy0EnOnQ9d/gOJ87Jlt1v3QT4sDe6qm6LkavfbCLV3Je/PwC0aNbd5w7V7bR5FIpi0ED31WT2Y4dO+6csCu7wgODQjBu/Cfo+UZP/DRt+uUqBTKjhf6bTWAb+kD/xSbIDpjNV1XZeTMsCacgn3Os+EU8fBaaVyPBtQqAafUR/P2GoU6vx8TPPseokR9eLnVhMJoxbux4dO/WA7l5+c4Ju1KhYPLX3yBxSzLefusdeLjPu6rvp6w3QT9zp5oZqHk1CtZT50FVr09PmMMywDb1AddhIUQ1XffqMirFJSXo1aMXPP/M77lUyuDH76fheeLGKCVZyqtN5PCwHz+eidGjP8K2bTtwIOMgnnv2OSxbsvTa6SOW3uCeAqbhAvDtwmBJygJV/ZOkNUA/dwfYRt7gX1oE8UD5d8T37ElD6yefwmudu+Dd/gPUR6XBcUrKbphMtZ8UWOOwC4IW06bNUMvqKfnY3V7riscfbY6EhITy7YXNBvOq42Dv8QX3qB9MS/YTg0ELL9Ub0M8x0I6OBdPAHZp+K2HLv1D+8yQJ3t6+anm6PXv3IjMzC7/+OhMvv/iyWn/IVgetRWsc9qjoGLVY/AVySSsuPq+6L22faUfO+r3X+Ss7xH0F4DuHgbnHG/qfkyCX0moFdSq7Xd2foOkeqWaw6ibGk2Py73fAjUYjPp04CaM+HH35d8UlFzB86Ah0fLmjeoV3GtiVXjc7U3bh+x+mkUj8iv89d85ctRREbt6Ndy9JRQyEUWvA3OUBzevEjz9SCFdNEa5TzklQaQpOJ1db4p8394cxMBV2k3jdv8kjwWevnr0RGXnFXVXW1j8jJ8Czbdpidx2kLld/dQHicmQcyMCUyVPxapeuCAgIVmtG6klknpy8HT2791TbIkZGRFaoTbmsM8LgkQK2sW9ZxL/sEGSt624CqW3ZiljoPosDc4uHWjfdsqdi+0aXLFmGLq90womT2erPSlVopX5oh/YdVOv+x7naX4CodtiV3GuloM/Afv0x4J2++GbyFPAaLbKzczB+3Hi8N/BdDH9/CPoR/13J1a6QZVEi/8Qs8N0Wq+mi2jHrYDvl2AWC6rsUa640duZeXAjmbi/oxsXBVnjjKs0sy2HH9u3E0L2mBqcJ8fHYvm0bZs2ai86duqjGbvPmpDr5TnWSLlBpK0OCI2V7H3ufV5mVX3yIVh6riXnOvwTdV5tUw8K3DiVX0/3qSllFdDzzBD77dCJGf/ChWp9RefxwxEjyOAozZ87C6TN1t5fBoWBXJSurNYeheXUxSt3mQhi2BuLBAthlx9yEXK+mVtAR3zwDXIdgsA1JEDpmA8QjRU7z/RwP9j9lK7oE/ZdbwTbzAfuYDwxTkmHLo65NZWS3mCFuzYWWGA7mtrngX4kk1vyAuqXOmeSwsJdJhrgrB0L/VWotEv6lMJgC90O65NplO25G4sGz0H1OXMOGnuAe8YP+922wnb3klN/VwWEvk8RrYYo+BE3nSNUyCd2jYAo9AInRUJr/FfIz0P+YDLaFH5hGHtBNiIc17YxTd05xCtj/ku3MJZgC0qEhFp75jzuEN6JhDN4HqZj2elKkgGzZlwf9NOL+tQxQC4tqx62HZVMOZIvz1+R0Ktj/knSmlLgz6eCfDwXzXw/i3kTCMHMnrNn1Yy9krUNuNsO86Ti0H25QyxEyDQjk49dB3JQL2eQ69yycEvbL0BewMIUdBN8zAszdnuoymnb0Opg2HIXsAtWFrdnFMASkQHh1ObgHfcE29ib+eRzExHzy/Y0ud9I7NeyXob8kwBKfDe2YVWqjKu5BP2jaLYF+5lb1jqBscJ5VB/kSiV/WHiZWfC34lqFg7/IG19of+l+3wXrofJ3XbqGw1xYIehNsh4ph9NoN/tVFYO/zBddMqT8YCb17MsQ9p2HXEvDtDhSk2WTIfwgwrTwI7ZQN4F+IAPeQP1hiyYURK2CJzVIb6ioVdF1dLgX7FeolyBojxN1nof8tCXyXMHCNA9Rtgfwzi6CbvAHmFYchnSpV4ber7QfrQQIa+dx2swg7o3z20zAsTIUwaAU5YYMI4OTzPxIIYegKGMNIUJ7L0kYQFParZVcA0lkIPGegn5Wk1qTkmgUTeALJCeAPvlM4dN9tIADth2VrHoGIhywYCXQ1eBLYyWcSRTU/Rb6kh+0UC8umbBjmbScwR4N7Ivjyyck9HQph/Cry+dJgy2HI57LWjxOTwl7v0SdRnQRbXimM0enQ/bIJmjeiVN+Xa0qs5/1+xAf2Ad8rAtqJsdD/vIUEwBkwr8qCNfUPSMcZSPnkZGB1kDld2eP1BqeHfFFLriAc+VsW1h2FMC8/BqN3KnTfboQwkgSWTwWQOMMHXBNy1XkshHyWMGj6LIdh/jZYkk5ALhbo5hYKezXhT3xi6Q8O4pYTMAbvgm5aAoSuMdA8Gwm+VdhlF0LpIcXe5QW2CTkZui0E3z2MjEVXRo9//KyMXuTvOy0E23hB2d/fS6B+OAB884XqawvPL4O290rop2+GafVBiBkFdBMLhb32pVREsOWWwJJ8Aub1h2Hw20nGDugmJkDbfTUZa64aSk8lods/f09+7herxg0G3x0wRu6FOeEoxP2nydXlArnK0E7hFHYHVJl/T0Vhp6KisFNRUdipqG5a/wfOayihKWj+rAAAAABJRU5ErkJggg==)
Answer:From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,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)
For Triangle COQ
OC = 5cm ,
CQ = half of chord length = 4cm
From Pythagoras theorem
![](data:image/png;base64,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)
![](data:image/png;base64,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)
OQ = 3cm
For triangle POA
OA = 5cm, AP = half of chord length = 3cm
From Pythagoras theorem
![](data:image/png;base64,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)
![](data:image/png;base64,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)
OP = 4cm
From figure PQ = OP – OQ
Therefore PQ = 1cm
Question 7.AB and CD are two parallel chords of a circle which are on either sides of the centre. Such that AB = 10 cm and CD = 24 cm. Find the radius if the distance between AB and CD is 17 cm.
Answer:![](data:image/png;base64,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)
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAvCAMAAAD3nYnBAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmY6Oma2OpDbZgAAZgA6ZjoAZjo6ZrbbZrb/kDoAkGYAkGZmkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bpRHQVgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACHElEQVRYR+1Ya1eDMAxNp67TDXWiG9M6H3T0//9DSyEtr3paYNkH5dN2DjQ3N4+bFOD/mZ+BrwfG1vMfG3Nicb+D42If88lZ3pWry2MQm7O45jtUsD7vR+p0yHsYBDUE6GHINYT8iSgY6uVVW6oxSHS+SJh+qDAcTOYhD9U/4kcsjUHEoFIq352f2ANsPuRXb8QsQFbR4HJSpdTRsK3Q1QU5EeLmu8MDFAktEc5eoz9kiIsmLxoRcL2g3zLHgAnWfoFFkDMnGJKPCEbXZLD2q3SIdpXWtRLh/qDJIO0vkkFrcQmhKazaWs9kkPZ7WM/i2lReY+iaDNN+/LpDuvDMcJ4+LrnhoWsyUPvnxFCbLCPzzrUTodrvcdgDDQZ4kHr6vit5sCYlv91+rPbB2j8Zg+RbUM86H5xJb2Vn5UBinlaoJ2MwFdSmrc6O4LpGDBYhqzR0YH4a9KJq9W2rYzF0QAfnQxSGM8UiCoMnOlNrs+rq7VN8JPoSZCoGEIsdnB6b6SM5Y1GSF9mr+/1BHdhi/cmn7ABWs9SH7jXXuOHGaVZwFQ6/aLU7Y1s4JbVUjdHuCUBwhjHTtUANHjPDjAfRGtswQ+eZ5YJBtWZoHBtI06Hk3w1z2GSpZ/vG/GX3K/Idx+56thjodz1LRLbEhStumAxOvd9erHZ/kxe5brKX2P0BylsPM5arTP+6yB0IlHdBtbZvwN4FzcLx3z3kB1ajKQ9strxWAAAAAElFTkSuQmCC)
In the figure, AB = 10cm = l
.....(1)
And CD = 24cm
.....(2)
As Distance between the two chords = PQ = 17cm
We have OP + OQ = 17....(3)
Equating equations 1 and 2 we get,
![](data:image/png;base64,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)
From identity
we get
(OP + OQ)(OP–OQ) = 119
Using equation 3 we get
OP – OQ = 7 ....(4)
Solving equation 3 and 4 we get
OP = 12cm
OQ = 5cm
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,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)
Using this in Triangle APO we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence the radius of the triangle is 14cm
Question 8.In the figure at right, AB and CD are two parallel chords of a circle with centre O and radius 5 cm. Such that AB = 8 cm and CD = 6 cm. If OP ⊥ AB and OQ ⊥ CD determine the length PQ.
![](data:image/png;base64,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)
Answer:From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
![](data:image/png;base64,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)
In triangle APO
![](data:image/png;base64,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)
OP = 3cm.
In triangle COQ
![](data:image/png;base64,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)
OQ = 4cm
From figure PQ = OP + OQ = 7cm.
Hence PQ = 7cm.
Question 9.Find the value of x in the following figures.
![](data:image/png;base64,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)
Answer:Triangle AOC is a right angled triangle at O and sides AO = OC.
Hence ∠OAC = ∠OCA = y
As the sum of all angles = 180°
90 + 2y = 180
y = 45°
∠OAC = 45°
Similarly in triangle AOB
Sides AO = OB and hence
∠OBA = ∠OAB = a
As sum of all angles of a triangle = 180°
120 + 2a = 180
a = 30°
In the figure,
∠BAC = ∠CAO + ∠BAO = 45 + 30 = 75°
Question 10.Find the value of x in the following figures.
![](data:image/png;base64,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)
Answer:According to theorem, angle subtended by the diameter on the circle is 90°.
∠ACB = 90° ; ∠ABC = 35°
As the sum of all angles = 180°
∠CAB = 180– 35 = 145°
Question 11.Find the value of x in the following figures.
![](data:image/png;base64,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)
Answer:According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
∠AOB = 2 × 25 = 50°
∠AOC = 2 × 30 = 60°
At centre O
∠AOB + ∠AOC + x° = 360°
x = 360–60–50 = 250°
Question 12.Find the value of x in the following figures.
![](data:image/png;base64,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)
Answer:According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
Hence Angle by chord AC on the circumference =
(angle subtended at centre)
∠ABC =
= 65°
Question 13.Find the value of x in the following figures.
![](data:image/png;base64,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)
Answer:As the sum of all angles of a triangle is 180°
Angles of triangle DBC are 90°, 50° and x°
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 14.Find the value of x in the following figures.
![](data:image/png;base64,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)
Answer:According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
Here angle subtended by a chord at circumference = 48°
Hence angle subtended at centre = 2x 48°
Therefore x = 96°
Question 15.In the figure at right, AB and CD are straight lines through the centre O of a circle. If ∠AOC = 98° and ∠CDE = 35°
find (i) ∠DCE (ii) ∠ABC
![](data:image/png;base64,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)
Answer:According to theorem the angle subtended by a diameter on the circumference of a circle is 90°
Hence ∠CED = 90°
As the sum of all angles of a triangle CDE = 180°
∠DCE = 180 – 90 – 35 = 55°
As the sum of all angles of triangle OCB = 180°
As AB is a straight line ∠COB = 180–98 = 82°
Hence ∠ABC = 180 – 82 – 55 = 43°
Question 16.In the figure at left, PQ is a diameter of a circle with centre O. If ∠PQR = 55°, ∠SPR = 25°and ∠PQM = 50°.
Find (i) ∠QPR, (ii) ∠QPM and (iii) ∠PRS.
![](data:image/png;base64,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)
Answer:According to theorem the angle subtended by a diameter on the circumference of a circle is 90°
∠PRQ = 90°
As the sum of all angles of a triangle is 180°
In triangle PRQ,
∠QPR = 180 – 55 – 90 = 35°
∠PMQ = 90°
As sum of all angles of triangle = 180°
In triangle PQM,
∠QPM = 180– 90– 50 = 40°
Triangle formed by POS is an isosceles triangle as two of sides are radius OP and OS.
Hence ∠OPS = ∠OSP = 35 + 25 = 60°
As sum of all angles of a triangle is 180°.
∠POS = 180 –60–60 = 60°
According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
Angle subtended by chord SP on centre = 60°
Hence angle subtended by chord SP on circumference is ![](data:image/png;base64,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)
Hence ∠PRS = 30°
Question 17.In the figure at right, ABCD is a cyclic quadrilateral whose diagonals intersect at P such that ∠DBC = 30°and ∠BAC = 50°.
Find (i) ∠BCD (ii) ∠CAD
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKkAAACxCAIAAAAwMgqOAAAAAXNSR0IArs4c6QAAIjFJREFUeF7tnQt0FdW5x4HwCCQkcK9gLSGkhWIRhBTaq4BCqHpRBEnVunxgweq6Km01eq+lIhVYF6nYhaAtD1dpiddXS7UCVoSKJSIIlqIIkSig5CUoUMg5SSAJSbi/ORvHcc6ZmT1zZubMSXLWWWednOy9Z+/vv/f32Pv7vt3+zJkz7dperZICHVrlqNsGrVCgDfvWOw/asG/DvvVSoPWOvG3dt2Hfeing8sgrystnzZw5IOcbvPMnXbN1y1b+dPkZLjXXtu5dImSkGZAeN2Zs8Z7i1a+sPVB68IEZM+75yU/69s128xnutdWGvWu0LN6zZ+qUKVddPQHgh1x4Ie2OvmT0qEtGZ2e3Ye8akQPa0K8XLMjMzJw3f762f0OGXAj8wexx27p3BxcWPQz/xptvAn5ti3fefZfuF3ee50Yrbdi7QcV27ZDxNMQqd6c5X1ppw95NMmd8ddG72bQHbbVh7w5R+0YUOji/trmnli3X/eLOw1xqpQ17dwiJSs8bdQ+8RYuY9eFwSCj8wXy1bzvDdREY8P7j8y+IBjHuUfRcbNz1plo+9s2l4abSUOQzDPmaI99N6NghJ6NDjqKrpyhfMlJyMvl0ne5BaLClYX+mqr5x15HTRZVNu46AceP7R1Uqdxqbxff2Pbqk5PY2IT0VaYQCp9+sVIt1HNZLmQe5vTvlZXXM7U0jQQAvzj60BOzB+HRRReRd2VymrOkO/TI65vYCKj7BqVNeX8dkotnIfDqqnUy0zySgWd7JyxWSGHvwqC/8oLGoQixulnVHBQxv16XKV3iuYAzMg875A7pMG8w8czzDElIx+bAXkDesPsASF3QHbz4TQj64QsPqj0/ufu9kz/dqT+/L6veLbrePTpZJkDTYs+CA/NTid1XIE77U6j7/ILxvQ3jf+rpPS5pXZbT/t6Zz+t/YuTCDGdm1YDjTMeDiIAmwZ22Bev3Te9tndumc3z+1YERiFxZgA3lt+bbTocpOmVnds78f/u3OlE7p7a8KpQ0c3eucn9Yt3gkzOBOq7zL1AiZoPNqGp8ws0NiD+qk52xCrrKRuc0aykhKlYDfVhYG8WoH87ab66tTeF/QcekNa9sjUcwdX/vyh8OtvfPP5wvCxdSf2rDp/+nYAg0shlU7O2QaXQhFJjbABT4F00HhAsYdwtQVFgnBd54xM1NJpCFWA94ndq+qO7IW4Gd8a333g+IyBV6aknrX4D8979Fjhs998bmXaRd+DExx87ocDp2/rnPmlWaGbvrABByB5VCVw2KvESiDqoCgYu+DqrO+MCOQ6DE68tLpyxqysBfN6Xpcv/lXy+KCvXT4XlqArGYRBRU+gAGGPmV5bsKlhzccJQR2uDj8H7+p96wVXT+s3sueFN8DVYy672nd2fHLLbedMm3LerF+oBcpfvJ3v2df/PmYV7QxIL7wy4ZpgILBHOqLAn5q7DbmetjjPT9EIV68t2wZjD+/fAGBp2RezvmHsWr4dDWRdyYef3Dwt44rLsh57RPtfpMNnG2cPur/EhEur4qzr7JGYA4nSYOhh4rFnNdRM24BohxYodB7JNl2zX5pnR/amdOnefeCVEcb+pSA36UZTuPqTW6ZR4JvPFaZkdNeWZCbtWzpywI83GHELtTBqYN3idwE+vXB8orSZRGLPcldI8MS7MHl/eCBSPCLLzwpypLiA3NaE2z/putOVn57/5us64EUjHy29GEnR+9L7LdtExtVMW48Vk3rvcCa9/wwgYdizPVedvwb4GTYmkCWlHBcwMc8ctKladKmDvh2z+uGNsxEiA27/m2TjrH4WQEIYQGKwZ8C19xVxONZ99WSPVB5hnkUWumJwC/MM9c1ckJsD9vmTS488ubTfsieQ9EYleWL5S3cMum+vagdaTgKVAfgp9ZRe4bvh56v5RF148upj7RbWzn7bi+ee+qz40OsPf7jkoj3z++xd+O2KVwpCH73myoOOv/jy7v6D+TRvrfFUiEc7eCgEgSyhsX+CRK502LIRX9e9yue7r77GXQVHmOPCPMMiR5CbmGeWC1FesY/Z1MHnru+U2Tdr4iK7D0Ltrc5fC/+HHfqwb20b+xHDckOhkDoq/NFwTsJVzXKc2Dbo8zjDuMXnnZlnlv2MLtBQ+emBSdch4Nm4lan+rx0rju1YITZ37b7g/6hBuBih/3tu61pyhugC/fvlrPvrq/xeXlY2eeIk/tyze7d5O3Uri2FocPv4GRpc/dg/frd/xRWCq5f9+cfH3/8TnNbBQGSqNIbC+yZey5svMuUpQw/pW31VuWR5XTFIVD31NcgF0Zy1IFnLtrwHZsCuqqoSD+DL8KHDfjp9usnzau7dxEgYj2SfYhZDgqqCHHHOdwcy1UEHPr5p6ge5F9VXVNqqy6RkgtqqoivsCtHMO2Ab++VLl7HWtY0CPPAbPSaeKcxqZk2jr0FKVhJr/fPNC1lV8dDUVt2KB2YC/Km9JbZqUZg+w5Ds1tKVF8wyzjVj0gfb/vnFxXuumjBBJ8a0GoD2X+xdcO6evnK8rfMrBDkiE42pZNEFn756f3NdmAMSzscwmtkzsdwysytijcpzVHPiL2vYrjcy5U0exJaR2CSO5wXRIB0EhIzxtGNY19bchMNHS3fYgI4TiDbtrnhhnqmCXJhn3gly84GH/rYRi+6zJ5bYoo9aGGEPo6opc8GO9W712+P5qHg69r7lrS3MBgSBjkbywAMwMAuLXAhyP7l6TGhh8rB6GL4z4EUtJjFjiacFta5H8Nuz8Yg5gr2rIeZ8nzrlVj7JNqCNNMbtgl16E1av7rMKxijcYCxPzzzhe1GNclTz0dgrOmX1+dYrL8XzRLubu+bPwmut5rYNOIFx8BFPr75SV35iYtGx6IV1x+uxRx/Nu3QMb37XNiImaUz7xGfzTH5oakkHFp3RU2D4cDIXZZYJYR2MlCqy654wM13OIBY6qQbY2NFOJTZwqn+wVjc9Jd1gXJvOcTRkeVRjq+3iX2X1ufrxaDceW41oC1syVFsty2Iv0yhbtuG8VbjSwpcEVwd11Q2Go1K2Wn3T0mU6rCujdb5zUD26Cm48HVIzHGzumjwdnR8n4IyiG+Lf9HUNe05jq3KfaR7W0O6BjrWfvSMEOadnHJ0FRJCbwxntfBc//FiqR95aaO7GY/cp0JkF1lxV32PXrXEe+buG/eGfzw2lvtHY9RPhBhNxixglf45plwTulo/pfBf/I/AOOvCH8TJuPLaeBfwnclaw7ln9tirqCztTE3S1OH98f9aE4mXXumLRutIl+UZcseiMHofVGufmbsyWGzaVx38ObntfL3qiKe6nc7c1px98uuvYP58ZEddM9L0yFl3Fzx/CotO627rYC2WDb5/7u3KcgOPoAdkhvuPexos9/IeTWSUGtjmc1mfwbTtr55Scctwb/yvidYnzXbTXpVs9QfYJxyHXX/i6QXaIL7IFOHjFiz2+ZjxbbDj8z8BuK0ekzS2py99WU3U6Ca7jwKJTgH9e727rgI5GVVB6+JcXS59mIXtk7TnkK3FhD8Nh/44J2NBV4Tw4zEzr1/m9yzKKjjXmba7eFWpykYiuN3Vs5TMc1eBg7+CoRr4zaLvsWmLryleRL4mrIy4eRLOwrSJfSy3pHHuV2+Nl21SvZLsQbpC5mSmlVyr5aoB/9aHTDvrkQxUsusOPLDjvoRkmXpdudYONDfzJ3GpN1w6+PZ0n92fPxwHnd449kTQqt9d1qEen9rsuy8j/eqcfbK8JoPgnqoZtnJ7XTj7ntls9gkTbLDscxPVxMO3Rs9IWjxORDnbbd4g9bmUomax44WGNIQvD1z27cERaAMU/zneEU8HndeFUdgknXx5Vnz0PnPblq9gqCQSIXYQv+6r2KtoqrRZGvxAx8eIXdnA7R2HP74j/TZd2D474x6Iru/seLLp+y3/jbODOaqHxESzgrK5MLWUR9ss4WbBJprBaxsm6VzJavVmJliHzpLxeHeH/ARH/QrHvt+zJmOFUMsNxVoZdbYJ8ndWVrAUcgGJL6XOC/cmCIixLrYN9hOcb5jHL6dahaEz3hIt/5Yxu49+x6Dpn9ZEkqFvFEPkEDkAltxqMbkdJ+DY2C6VP/hG2sVfSmr1/lFwY2mc01YfMY53Q/hD/i4Z2S5T1L5zvyJPgqUVnRHeIgz7knbYvnttt8TjCmQFIEn7b2KNP6ha95JMoVjCgS0LEP/lwSJDR+57paoIM+T67VZINPo92eNQecrqD54S8wm8Pe+YUM4v5paMI25bRen5Mqvkv/rHo4PZYdOfeM90tIB20g7ZP0h6UYgd15at0mzNKfunbw574WWXrPlYCyc49ZPOW+in+UexL71IUe98sOiOcxOau1xof9p780reBPep9tKSXn5LaklrxP21nrUeb/9oEGc766WItNndJ6OK1yKfDePWz9GXO92xgz6InYj46flZsVqf2tp09TIh/9n3Z/S092ewioUVTbN5h0eUs99uiMxoIm7sebexrnygUfsCypKcs9mzkcWZgkiDDmYuOKv5z3wgXHW207K58AYBHsWcPJyGKfcx+IvI93dxVH8rSByzzawIoLIt9XeEHpDS1FVoliZMq/se9Vb34gMOjaN2zsOhIeYhFR8pDyW74UAw/VTZ3Pd3gE6MAJsDiwMV8ULLYo+EbAc9ONceU8dBOFf/37T4Zv/hXFPsZs8h8l0CLzogaeDJ6t7Gvfaiy9K0OdqWwR3FAfTBZ9M4Yvo5Aroj/s5nvLv++Rz5Y8Uxx6roSoynTBzL3AZk5/FLYs+jR8uJ3CLfsNOKf3V+KORP/qvNdwi06o5Gyucu/fND4MPaAzAXsCQYwWfTYrC5GXOD6Afx553R0IP69dr6znLuWBdjcRT56vcGnSn2AM+mS9bpn7pAK3jz3S0qXr1wCa0kC8wKI/9Uj02cPSrUl/n1wvotzXKI6S98fkQ9kAGey9K2x5wIicUuU0cg92qecM6jryxenS1r//jjfuYO9L5u7dNWS7Vtjz8ThDiKTYbNNLcSY6y+OfWXEv5/Od/GPUaRw9Yftd8ofwNI16rMF9uwPmGv48dPCvAVL8e+z850r4yVM0Qd1j67C9oHPyJfLAnsWPbsEJhq+RwxfS2IT8e+/850r2MMmCU92pSnzRpTrATO7cJ1bzGIW2GPZcwOZKcNXPAUwW70eiVb8i7OfRDnfxT9SyOW1G4/aSeAzOtexxL7S3eSn8RBOFf8560MEfiTK+S6eIYi6mMS4O/jD9oHPSOSbYY+wx0hIMb3usanuy/yq8RPFsgVV/L8zZ3qinO8sOylTwKMYzehHo6cDYkyRb4Y9lwrTlvm6r/t8L8fSMqN1qwzif+WhjaNe2brwzp/cdVLWYcStp7vVDth7FKOp66HQ1bjMN7rn5tgfxbJ3a7RutaM63+XfcgfWP7u/Hrl+uNXhmO34aumNzbK/7osqzBm+MqHqfeX5Wuc7nfj3FCrXG/c0RlPXW+4Dt73uyeTdIcdisxafc+GJ5sMr2vlOiH8+v/NGuLCswYc+uPgIT2M0tf0kb73tda9cS2lq4LlICMumjJzvEP/Af++ALqR94Ozfsp3gFPA6RlMdKcwbdc+GvBdGYYrVum8IVbpyeG8JCT5YlWeOTCzbf/7QoQNyviHe+ZOuqSgvp+7ioUrah+QS/17HaKok7Zjbm+/RVr6FfW95kREOaC4e4BrNAC4pwvnuP375+IHSg9zUwTUdfHn62WeL9+z50ZSzcdTEfcIA0PuE9W85mYJQwOsYTTFGkYotOkDfEHs2BAjtDAKBOKrhdirV+S4z82yvmAFXXT2Bdb91y1bRTwQ/cZ9JJP6V81yPYzQFZbDXokW+2bpHRzDHXuQTSOni4RQRzndE1cR0vsvOztb1MLnEP/G5/mzuxszCaIi9TA4PGD6k947nq853Oh+s8vIKsdzJ8kta3yEXDtHNgGQR//7EaAq2H23mGWJPUfNje6/FgYnzHcCPGzN26pQpfbOzn372GW36drVXqvhn8yfI4t+HGE2FN+f2tiHvZaBtqKrA4VympIMyZXf9zCjzndD1eJO3H9XPqHEh/vH/J+4nsNa/PzGaMUlk7bdjApt3Sj5ndDD8+DPfCfEPD8D6L9h90sEU9LqKPzGa7mPvEV2E853RJUWhUJi3rUcL8c/ShwEEbfPftxhNt3l+CJ7vpocuiKrOd9GKPWKe/Rxset58sQW/EP9EfAZQ/LP0vT7L5zSPGGodxeLk+RXuKvnmzndodkLMi7ct7FXrP4DiH3XP6xhNezaeXcrGX17Zsfc4850Q/xwABkr8+xaj6ea6j5lS0dkkEEc1/mS+E0kfnzhQHxzx71uMphaduHg+e1LyqVbM54Rwvuv72CP+ZL4TOZ+x+wMi/iNuPB4m4BMBNm6ue2dLPLqWmvnOzzwJIuczUiAI1r9IwOepxhct8p2ve5EpUDK9lsksSWDmOzXnc8LFv58xmioWzrHX5k13zAC4pCjhme8CIv59i9G0xp6Te6N4DsdI6ypi0bFxm/DMd/QqCOLf083d5tKQDZ4v46kXD8NXne8CkiBDK/4TcuWDp567hFpwnGND1zM/xjXKmy7DFbTOd/4o9jK9SviVD77FaApqGMp7vDSjdwFlKChTJmiZ77R9TuCVDxE3Hk+uWCC9enRArXUMthGW5nnTTWaAcL4LVOY7XW8TdeOTSMDneqp1wb9tyHsRitVUahh6YZk3PSb8Ouc7GSaRkDIJufHJoxhNEVsnvHW1L7N1T+h2k80rWMxxMne+SwjGJg9NiPj3IkZTJNi0se4jM6UXoTlG1CGUMPVcGykVjZzvgga5rj8+X/ngRYwmHrpk2I2ms6mfrkEcl9pKSqrs4X3wM9+ZTEEfrnx4atnyWTNn4n3qhaWH62W0gWem54t1j37oyro0cb5zpX2vG/H0ygcgf3vrltGjLwH+mvozrt+jybqPmTXHYt1D05hhfLbyprvlfOc1wObte3flA/5IuKUQZ5Kd3Rd/NCXVunuX6QFfJIGGXtGzXvcmmXqoLBOJZ+58l1g47T7d7o1PLOhfL1jw2qvrtA8CaX4kskD9EdSL9xTjdc4M4I2VT9o6t+7RFG75sbNlnTF9hSevrp76WnSRmrK398zvY16X/x5/8eXd/Qcf/cP/WZZMrgKbjpzOXHti2MbQe1WNRj1fvnTZ5ImTfnTLLf375Tz26KOiWHlZWd6lY7a8tYVf+Je2Lv9S/9y78NvH3/+TKzQBvtDY2E1Z7O1g5cfMySqTNz0ZM99JcgJL8c/iJnSI8AECRm+8+aZ1Xyx9JDrBBbwfmDGDta5lCax49ekuxmgqXhsGqTEtsDfJ1GPO8H1wvpPEyaNiWvEffeUDQM6bP199NGDzPRQKIQWGDDkbTEIo2R9feD5m99yK0RSpMY2SKFhhb5qcz4is7jrfIR1ZLiLOPlAvmSsfkOu8WeX0nIXOpxo9SESR+CX65VaMpkiNaZQty9p3o3N+/+irFs3zprvofIfhy3JhrQB/oIBXO2Ny5QOov7buVUpOjSQJENNXy9uNRuRWjCbYA5/RU2SwH8CBXvTFO0ZRGYcffrR24zuEU5k734GoWBMKgb4Qh9E6MFIT4NGE7cbi+DlRjG58QtIj73+zZAnBJIxRoK4yMCiQkWm4OYbTfpzemxzhsD1jkiLPGntRWZexwyiNLkc1jctLzz18Y+euXzenPutYfQsSQBSSaIwafQl4E2Mrqt94001PLV9ObpWrJkzwE067zzK58YmJK4S94PYV5Wez2zLe6Ohx9bliczeedMUic77ZxQcyhgR2QtWwr9hpGHiYebq6Ndv/gUV36Je/orCuvK6kMHV0P2LzPPTgg+LH4UOHrfvrq+J7VeQl088glFm0v67dS8en/rNm89btql3HcMQQGOBPp08Xg8L2w9gz6nPjqRBEDn0Uw8CWHCb2OW+Twu1kGqp/ef+xdgubDobUwtHYn9pb8kHuRRUPzKTM6feOUL52tn5yqNWxbhk5YKuDF7R44bnnRRn+pTN/ZfoZkDIvf9qQXriD4eSNzQNsBrJn9251HmP0Y/rzI5/mHf7k2esqXilwNijAAoK6lcXxYk/9f2X+VsVSTEntum8MhUvGXLFv4rV8EQ+jMFWMHsywmf6sBggk1jqTgO8qjZgc/NfZsINQiz2fQX/58LsLXli7Iwb1Gaw6UpPefr554YdLLnI2nFOLdprQX7Qpte4pV3PvpuP9fifqRG/qgTrYq8BTpvlEneW8oxgzQLA+VjxfVN6e7NgztBMNzZPfrob/IwUc4vdZMWusvqrcQXXAirkhq23KWtcT2ofJfWvCotNdO4unAGfGljfyogOTMYVTLFs6sF1FLCHlnd34pO2q4xhNcZ9hasEI84HLYo+7vnrDrjZvunC+IxdStEXHnqDllax0DiW/b99sWzpwQrB09lBbNz5FP8JZjKbJVfXaR8hiTx2uyMNeBE41b7rqfJdxxWW2SKM16Dm1xApi9Sv73pGdELH3eeNNN9tqM7CFJW980vZfMc2LKninpVxq18oXt1bL3FzcXpH50q8TOSvYHD55+146dN7wefsnXU9UjdEllCfnbCOyJ6PoBl3zbHRgrwO2OLucN/8RkS8JyNn/wo5HBGDl33n3XdL9SoKC5HqZ9s/aNYdPLxraja1Akx7XFhTVPfHlNcZN/Wu7rbosfbhswuqaaes5v+lZeoc1UWzpEdgMaHCH1sw6sGwyFl3pnT8zqY56aHR6iE6HchfTupXUgW11OziFZ+89Kax/oy6hoKGfC9sMfRnruvKOBw+9/rDkEGRMO7UpWT1frYACuf9/J5RcPlpr0cXsGSVNTHzJwbS8Ylj/4uwfQ0A3OsjF0mJ3RPs7wO9fcYUkHcTUYdLIlLch7wUP6TZnZP22Q6cPH28eU1n+ym3lL95+5K3HeePFpY0pEaqm+U2q1kypJZYwuvIBGY+O1nX2SJ2PjXyMJpK+/um9qQXDY2bXiaalPXkv6h+eOK/D4NSO1/dgt5kgEoI0yKQu8quKF96GXfd/t+POnu0fTlEuiDh3MD/6cI9aEk0VVfyT/YUwIHqOegT2yOlo5Ip/ldXn6sd7DtVrTrrxCknfY9etHmLPkW7NbRsyNv1Qd0akTIUjSkIG4WpoNC2YDZ0y+4pjSlK2kIrZ3WRdSTQD5pScmltSN7VfZ6IA0KNhk2mL86L7D3PtkJqRNXGRydDwyQx955n0leNlNHzRjtS6xyQrLy/XKt6h3GeYXNE6vFHnxLRgYwD7UJkckdwykclRrVYR921FNjQyW8+0IN6b6z6+1VC1/vq/ZL53a0ynyn/tWHFsx4rzp283wT6ctwqpkbnr7G0CMgvAGnuRuBgPJOxvtUXEeXjcn23NMqPe4JCKvJCZFiL9KMFAxIQokyMzWS9I05GClE+TNh5+6u/vfH1+PnGA0YRikRz4w/gBP95gxCA5rq3+wdpoTmw+A6yxFw7FYC/OodUX0gU3zpjySWbSyZSJMIYwGZuVyVEPz4gIlPIvpz+ZnL9QJpJ7WiD+87fVvHmsURX/Ovp8tPTic753x79/L4bVznIX+y7dV0+WoapaxgJ7UGe3la0Y/E+Ew6H6Eo/EJSi9ULnU2+dX9LSIiBVFoIhXhDFkIT7OTo7ILe2pvQfLxBT4PBb1cWR7JuufEP+6PlT+9b7ToYpv3PJidN/YC0IDc7AIzbCH2+NcPOHqCfD8TZvfjHY0M1L6EkU78VxhaiqToy4cESgVMtMiIGYICZ+ZAcILiNMglZIndq/69NX7hzyoD5ETwjdtUR6mnV2ym2GP49STS5YwA1j3RvlrUTHQMB1MOrsddaW8dlpYmiGJsk4R//B/pIC4+k8MnOlbsuiC7OtWiGBN8YL1VuU+w9U28kq3loyG2OMgi6QXRdl73/n+rpjUdyxsXMHSrUaCZp3GFP8Hfv+f+O2fd/lcddRC5cKgt7zOLCahYmPPcQsekhyuUweRz3d4vhGhhZLpis7vFpYutpNA61Qn/tk8De9bP+D2v4nRCYHb/eVrHG+exsBeRBCq99DwnamArmdCUKFuwHlix/y5CEWQmvLBOhXiH86/emR6p8PbDz73w4HTt2HcImeRtnEq2nrsVVYvDHpEvrh9DkXPZOnD+elKc1W9/IZikEB0vy8uWqdC/NNF4O/y+yFfu3xuj+zJUJtfWGyS27c2eL4DYnCQgN7hwMp08KykruLAOj3V2C6/tG9xbfrGqv/u3bVdz2evYd8+fi5rvbcjT2hhb3SZekFCLH75fga2pKV1Ss87tEs/d+Z/xSPm1eG7ib2qgLRUvS+Bk0ZMi5rN2+s2fdDroml3b16u3gTL72y5IqNj3hNo1meZQ35bZXAfkPHOttVmW2EoIJymVM9r3NtFiA8BTLi3W0Z6RNPQtu+G5dyH4cP2MT9iJuqxrN5WICYFICbGlFaeEtcnQvlHRfbaw2HDLJiGJPVoSREJhvOQzv3Io2e1+GYhI8TUhjgS1iNCWXgR3kUMkzZliyRB3F/3Ypax+tlrFDu+bUs5HgoIU163cSuE/Yhhubz5wq6rTFi/rhteYS88O9rgjwd16mqB15ryr61bRy4PDln4ZB6wK+PkQZL8wVkxHEZx025T/ZxRDwdtwep1frciZlll8kSzE9vr4BFerXsxDcXqF6pfdOIWJ1O11dSBXJySsGuLG5Zu8w4XOqJZBJMnoIXAJmdU8RZ7VfYL+NFUnfWytdWCUJDLaJfsjy+8ILKQcM5CJBPw33mXkxgml/d2TEAS506dJyt+PvHsQrfsecDJiHIyu+bjmO4YwndSSwEYAGlptK6U8vTxD3v6xKZvdf5aFMC0witb1YmfJB6KET9tPXnr0wvHOz6ZlXwWxfzg+Wpv8OcXHiYYLW3iXwcSBFFP53wAXtHGbMXhys8pk5KwtdqCTUQPIc/SFo9r4/+JIkgCsBfTAm+fmmkbAB7+ZpIDzpXZFuRGkIPQAfiJyJEPqXFlRAnDnt5z5K+EkL1ZmXrvcEI8WxsDAG8lBu+Jd0logv7rzOcunkmQSOxFvwlAhAR88UfBiYdYLtaF7WHIAT+T3oF7tSs9STz2ggGgAWDYsAK6LR7Xsk0Aldth7qLu+L/c1XkTCOxFb4TkI2ofHbDbnFEJJIorqyq6EVA/OedtlNwO/TKCoOUECHtBLEwdhRmG6lvSDIC3n1r8LtKNAfqv0xlN5cBhT0dVSrWAGSDWurh7BLlOmsLgqLRBxF7MUzEDYANK7pbJ/SFccpmCiDA6D4fn9oKgoS4oHFzsVU4FBeGW5PBHTLJuMIKDs3Si2SlTlg4za5myHYf1AnWfrXZ5ZSUJsBeDYa+7bvFOmCeCADbArifv4EwCIMdsU95rPmahc/YK5AFnVEmDvSoIVBLzi5gEkDhRRgHiHN4uIFf7E6hJacIGkgx7dSTadcaPiIPIJMjipm+v5wF4c6s4kTFkDUUSJR3kKg2TFXvtdI6knlWQELf3wnLZHSKTM58dcjLj3ylC3DSXhvjkjknle5lyNziynEfA2wPO2FvgujcaEvOgaddR1qXyGVmU6mxQAPvilkB8CKLZAwtavfOdmSSUDNQL0QhgpzClcnvzmbx4a+nWEta9ydSOwBliHhAjzNpVc7ob3e+t3hPPzIBndOjRBaRT+JKTIa8/J0vJFo59ssCQkH766reTkBG2PdSIAm3Yt9650YZ9G/atlwKtd+T/D82Hp7lMpMvOAAAAAElFTkSuQmCC)
Answer:As ABCD is a cyclic quadrilaterals, sum of all angles = 360°
As the angle subtended by chord BC on circumference = 50°
Sum of all angles of triangle BCD = 180°
Hence
BCD = 180– 30–50 = 100°
As the angle subtended by chord DC on circumference = 30°
Therefore
CAD = 50°
Question 18.In the figure at left ,ABCD is a cyclic quadrilateral in which AB || DC. If ∠BAD = 100°
find (i) ∠BCD (ii) ∠ADC (iii) ∠ABC.
![](data:image/png;base64,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)
Answer:As ABCD is a cyclic quadrilaterals, sum of all angles = 360°
As AB is parallel to CD
∠ADC = 180 – ∠DAB = 80° as they are interior angles on the same side.
As angle subtended by chord BD on circumference = 100°
Therefore ∠BCD = 100°
As ∠BCD and ∠ABC are interior angles
∠ABC = 180 – ∠BCD = 80°
Question 19.In the figure at right, ABCD is a cyclic quadrilateral in which ∠BCD = 100°and ∠ABD = 50°. Find ∠ADB
![](data:image/png;base64,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)
Answer:As ABCD is a cyclic quadrilaterals, sum of all angles = 360°
As angle subtended by chord BD on circumference = 100°
Therefore ∠DAB = 100°
As sum of all angles of triangle ADB is 180°
Hence ∠ADB = 180–50–100 = 30°
Question 20.In the figure at left, O is the centre of the circle, ∠AOC = 100° and side AB is produced to D.
Find (i) ∠CBD (ii) ∠ABC
![](data:image/png;base64,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)
Answer:As the angle subtended by chord AC on center O is 100°.
Angle subtended by chord AC on major segment = 50°
As the sum of angle subtended by chord AC on major and minor segments is 180°
Hence ∠ABC = 180–50 = 130°
As line AD is straight
Hence ∠CBD = 180 – 130 = 50°
The radius of a circle is 15 cm and the length of one of its chord is 18 cm. Find the distance of the chord from the centre.
Answer:
The figure is attached above
A is the centre of the circle.
AB is the radius = 15 cm
CD is the chord = 18 cm.
We need to find the distance of the chord from the centre i.e. AE
In this circle, we draw the perpendicular
We know that perpendicular drawn from the centre to the chord, will bisect the chord, such that CE = ED = = 9 cm
Now,
In ΔAEC,
Applying Pythagoras theorem,
AC2 = AE2 + EC2
⇒AE2 = AC2 – EC2
⇒ AE2 = (15cm)2–(9cm)2
⇒ AE2 = 225 – 81 = 144
⇒ AE = √144
⇒ AE = 12 cm
∴The chord is 12cm away from the center of the circle.
Question 2.
The radius of a circles 17 cm and the length of one of its chord is 16 cm. Find the distance of the chord from the centre.
Answer:
In the given figure:
Radius AB = 17cm
Chord CD = 17cm
Now we draw a perpendicular bisector of CD from A which is AB
Hence CE = DE = 8cm
We have to find the distance of chord from centre i.e. AE
In the triangle ACE
AC2 = AE2 + EC2
Therefore AE2 = 172– 82
AE2 = 225
AE = 15cm
Hence the distance of chord from the centre is 15cm
Question 3.
A chord of length 20 cm is drawn at a distance of 24 cm from the centre of a circle. Find the radius of the circle.
Answer:
The figure is given below:
CD = 20cm = length of chord
AE = 24cm = distance of chord from centre
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
From here we get,
r2 = 100 + 576 = 676
r = 26cm
Hence radius is 26cm.
Question 4.
A chord is 8 cm away from the centre of a circle of radius 17 cm. Find the length of the chord.
Answer:
The figure is given below:
AE = 8cm
AB = AC = 17cm
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
d = 8cm
r = 17cm
Hence the length of the chord is 30cm.
Question 5.
Find the length of a chord which is at a distance of 15 cm from the centre of a circle of radius 25 cm.
Answer:
The figure is given below:
AE = 15cm
AB = AC = 25cm
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
Here d = 15cm and r = 25cm
l = 40cm
Hence the length of the chord is 40cm.
Question 6.
In the figure at right, AB and CD are two parallel chords of a circle with centre O and radius 5 cm such that AB = 6 cm and CD = 8 cm. If OP ⊥ AB and CD ⊥ OQ determine the length of PQ.
Answer:
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
For Triangle COQ
OC = 5cm ,
CQ = half of chord length = 4cm
From Pythagoras theorem
OQ = 3cm
For triangle POA
OA = 5cm, AP = half of chord length = 3cm
From Pythagoras theorem
OP = 4cm
From figure PQ = OP – OQ
Therefore PQ = 1cm
Question 7.
AB and CD are two parallel chords of a circle which are on either sides of the centre. Such that AB = 10 cm and CD = 24 cm. Find the radius if the distance between AB and CD is 17 cm.
Answer:
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
In the figure, AB = 10cm = l
.....(1)
And CD = 24cm
.....(2)
As Distance between the two chords = PQ = 17cm
We have OP + OQ = 17....(3)
Equating equations 1 and 2 we get,
From identity we get
(OP + OQ)(OP–OQ) = 119
Using equation 3 we get
OP – OQ = 7 ....(4)
Solving equation 3 and 4 we get
OP = 12cm
OQ = 5cm
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
Using this in Triangle APO we get
Hence the radius of the triangle is 14cm
Question 8.
In the figure at right, AB and CD are two parallel chords of a circle with centre O and radius 5 cm. Such that AB = 8 cm and CD = 6 cm. If OP ⊥ AB and OQ ⊥ CD determine the length PQ.
Answer:
From Theorem: A circle with radius r and a chord of length l which is at distance d from the centre, follows the equation,
In triangle APO
OP = 3cm.
In triangle COQ
OQ = 4cm
From figure PQ = OP + OQ = 7cm.
Hence PQ = 7cm.
Question 9.
Find the value of x in the following figures.
Answer:
Triangle AOC is a right angled triangle at O and sides AO = OC.
Hence ∠OAC = ∠OCA = y
As the sum of all angles = 180°
90 + 2y = 180
y = 45°
∠OAC = 45°
Similarly in triangle AOB
Sides AO = OB and hence
∠OBA = ∠OAB = a
As sum of all angles of a triangle = 180°
120 + 2a = 180
a = 30°
In the figure,
∠BAC = ∠CAO + ∠BAO = 45 + 30 = 75°
Question 10.
Find the value of x in the following figures.
Answer:
According to theorem, angle subtended by the diameter on the circle is 90°.
∠ACB = 90° ; ∠ABC = 35°
As the sum of all angles = 180°
∠CAB = 180– 35 = 145°
Question 11.
Find the value of x in the following figures.
Answer:
According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
∠AOB = 2 × 25 = 50°
∠AOC = 2 × 30 = 60°
At centre O
∠AOB + ∠AOC + x° = 360°
x = 360–60–50 = 250°
Question 12.
Find the value of x in the following figures.
Answer:
According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
Hence Angle by chord AC on the circumference = (angle subtended at centre)
∠ABC = = 65°
Question 13.
Find the value of x in the following figures.
Answer:
As the sum of all angles of a triangle is 180°
Angles of triangle DBC are 90°, 50° and x°
Question 14.
Find the value of x in the following figures.
Answer:
According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
Here angle subtended by a chord at circumference = 48°
Hence angle subtended at centre = 2x 48°
Therefore x = 96°
Question 15.
In the figure at right, AB and CD are straight lines through the centre O of a circle. If ∠AOC = 98° and ∠CDE = 35°
find (i) ∠DCE (ii) ∠ABC
Answer:
According to theorem the angle subtended by a diameter on the circumference of a circle is 90°
Hence ∠CED = 90°
As the sum of all angles of a triangle CDE = 180°
∠DCE = 180 – 90 – 35 = 55°
As the sum of all angles of triangle OCB = 180°
As AB is a straight line ∠COB = 180–98 = 82°
Hence ∠ABC = 180 – 82 – 55 = 43°
Question 16.
In the figure at left, PQ is a diameter of a circle with centre O. If ∠PQR = 55°, ∠SPR = 25°and ∠PQM = 50°.
Find (i) ∠QPR, (ii) ∠QPM and (iii) ∠PRS.
Answer:
According to theorem the angle subtended by a diameter on the circumference of a circle is 90°
∠PRQ = 90°
As the sum of all angles of a triangle is 180°
In triangle PRQ,
∠QPR = 180 – 55 – 90 = 35°
∠PMQ = 90°
As sum of all angles of triangle = 180°
In triangle PQM,
∠QPM = 180– 90– 50 = 40°
Triangle formed by POS is an isosceles triangle as two of sides are radius OP and OS.
Hence ∠OPS = ∠OSP = 35 + 25 = 60°
As sum of all angles of a triangle is 180°.
∠POS = 180 –60–60 = 60°
According to theorem that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference.
Angle subtended by chord SP on centre = 60°
Hence angle subtended by chord SP on circumference is
Hence ∠PRS = 30°
Question 17.
In the figure at right, ABCD is a cyclic quadrilateral whose diagonals intersect at P such that ∠DBC = 30°and ∠BAC = 50°.
Find (i) ∠BCD (ii) ∠CAD
Answer:
As ABCD is a cyclic quadrilaterals, sum of all angles = 360°
As the angle subtended by chord BC on circumference = 50°
Sum of all angles of triangle BCD = 180°
Hence BCD = 180– 30–50 = 100°
As the angle subtended by chord DC on circumference = 30°
Therefore CAD = 50°
Question 18.
In the figure at left ,ABCD is a cyclic quadrilateral in which AB || DC. If ∠BAD = 100°
find (i) ∠BCD (ii) ∠ADC (iii) ∠ABC.
Answer:
As ABCD is a cyclic quadrilaterals, sum of all angles = 360°
As AB is parallel to CD
∠ADC = 180 – ∠DAB = 80° as they are interior angles on the same side.
As angle subtended by chord BD on circumference = 100°
Therefore ∠BCD = 100°
As ∠BCD and ∠ABC are interior angles
∠ABC = 180 – ∠BCD = 80°
Question 19.
In the figure at right, ABCD is a cyclic quadrilateral in which ∠BCD = 100°and ∠ABD = 50°. Find ∠ADB
Answer:
As ABCD is a cyclic quadrilaterals, sum of all angles = 360°
As angle subtended by chord BD on circumference = 100°
Therefore ∠DAB = 100°
As sum of all angles of triangle ADB is 180°
Hence ∠ADB = 180–50–100 = 30°
Question 20.
In the figure at left, O is the centre of the circle, ∠AOC = 100° and side AB is produced to D.
Find (i) ∠CBD (ii) ∠ABC
Answer:
As the angle subtended by chord AC on center O is 100°.
Angle subtended by chord AC on major segment = 50°
As the sum of angle subtended by chord AC on major and minor segments is 180°
Hence ∠ABC = 180–50 = 130°
As line AD is straight
Hence ∠CBD = 180 – 130 = 50°
Exercise 3.2
Question 1.O is the centre of the circle. AB is the chord and D is mid-point of AB. If the length of CD is 2cm and the length of chord is12 cm, what is the radius of the circle
![](data:image/png;base64,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)
A. 10cm
B. 12cm
C. 15cm
D. 18cm
Answer:In triangle AOD
AD = 6cm
AO = r
OD = OC–CD = r–2
Applying Pythagoras theorem we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
r = 10cm
Question 2.ABCD is a cyclic quadrilateral. Given that ∠ADB + ∠DAB = 120°and ∠ABC + ∠BDA = 145°. Find the value of ∠CDB
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)
A. 75°
B. 115°
C. 35°
D. 45°
Answer:As
ADB +
DAB = 120°
In triangle ABD as sum of all angles = 180°
ABD = 180–120 = 60°
As ABCD is cyclic, opposite angles have sum of 180°
Let
BAD = x°
Hence
BCD = 180–x
Let
BDC = a°
Let
DBC = z°;
ABC = 60 + z
Let
ADB = y°
As sum of angles of triangle BDC = 180°
180–x + a + z = 180
a = x–z ...(1)
According to question,
x + y = 120 ..(2)
and
60 + z + y = 145...(3)
Subtracting (3) from (2)
We get
x + y–60–z–y = –25
x–z = 35
Equating from (1) we get
a = 35°
hence
CDB = 35°
Question 3.In the given figure, AB is one of the diameters of the circle and OC is perpendicular to it through the center O. If AC is 7√2 cm, then what is the area of the circle in cm2
![](data:image/png;base64,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)
A. 24.5
B. 49
C. 98
D. 154
Answer:In triangle AOC AO = OC = radius of circle = r
Using Pythagoras theorem,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJUAAAAgCAMAAAAse8q7AAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjqQZrbbZrb/kDoAkDo6kDpmkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bN1O5hgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACCElEQVRYR+1XaXODIBSEpolp2torsUdsQ3MUUvj/v68POUQBlZmOk07Dh0Tji6y7+xZE6Dz+GwP7B4xvTu2h+d0b2l6sTw0W4GHzU0RFbkeh6vi47JuHLj5MyXYcW9GIIgTXA7FMIyfjgIrZRDw37MOylWSLAihaHQ0aaS1bV4tiFr4/f/pqXCBTOOe55G8wqrSWdapprM9pC60maxBFblFay+rqUlKAkDaROqlGCYyIHYTmpVIySmoPzrSWVdU8V5RIDGDpWhvxsoFf8RId8wkcSeD6O42tQMuKImoBXc0yJ33cedk10FZKyET7KKp1F8xQy8ZRmWrqWJfnLkJ7bEoSjEWhIz4zcGywZduo/GqXAKKM/32Qn+82F0qtXIPWHhlZtlju5utwy3pcedUaSW3n4yuWsvF7ZSbHbGFUpY1at5k7nyCAqrWIOaiUUvtDxY3NBVGYf6RxFTF08CE8bzgK6oyATgQcRN/VyYNfQaVEaUH2UVm320lFMdnY5aac2QQLuz2mYEf+96OqCTCmBvXwqsoFmQoyVqnSMCUZOvu1H5VJUYhT6zhRTLfqpMp/UaqTlBR1A8drVw+VX23c1IgIfKUE0PJUqFJWHJZhHN8btlEFqkO6cLPIuI+ZEKI9YTbgcpACEtjdVBYbbQzcaYxKleytIe8sdoc8GltpbxOjwTpP9GcZ+AGiHyubZ4VL4gAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
As area of a circle is given by
= ![](data:image/png;base64,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)
Question 4.In the given figure, AB is a diameter of the circle and points C and D are on the circumference such that ∠CAD = 30°and ∠CBA = 70° what is the measure of ACD?
![](data:image/png;base64,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)
A. 40°
B. 50°
C. 30°
D. 90°
Answer:
According to theorem the angle subtended by a diameter on the circumference of a circle is 90°
ACB = 90°
Angle subtended by chord AC on major segment =
ABC = 70°
Angle subtended on minor segment = 180–70 = 110° =
ADC
As sum of angles of triangle ADC = 180°
ACD = 180–30–110 = 40°
Question 5.Angle in a semi circle is
A. obtuse angle
B. right angle
C. an acute angle
D. supplementary
Answer:As a semicircle is formed by a diameter and the angle subtended on the circumference by the diameter is 90°.
Question 6.Angle in a minor segment is
A. an acute angle
B. an obtuse angle
C. a right angle
D. a reflexive angle
Answer:The angle formed by a chord in the major segment is acute whereas in the minor segment is obtuse and the sum of both angles is 180°.
Question 7.In a cyclic quadrilateral ABCD, ∠A = 5x, ∠C = 4x the value of x is
A. 12°
B. 20°
C. 48°
D. 36°
Answer:In a cyclic quadrilateral sum of opposite angles is 180°
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 8.Angle in a major segment is
A. an acute angle
B. an obtuse angle
C. a right triangle
D. a reflexive angle
Answer:The angle formed by a chord in the major segment is acute whereas in the minor segment is obtuse and the sum of both angles is 180°.
Question 9.If one angle of a cyclic quadrilateral is 70°, then the angle opposite to it is
A. 20°
B. 110°
C. 140°
D. 160°
Answer:As the sum of opposite angles in a cyclic quadrilateral is 180°
Let opposite angle be x°
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O is the centre of the circle. AB is the chord and D is mid-point of AB. If the length of CD is 2cm and the length of chord is12 cm, what is the radius of the circle
A. 10cm
B. 12cm
C. 15cm
D. 18cm
Answer:
In triangle AOD
AD = 6cm
AO = r
OD = OC–CD = r–2
Applying Pythagoras theorem we get
r = 10cm
Question 2.
ABCD is a cyclic quadrilateral. Given that ∠ADB + ∠DAB = 120°and ∠ABC + ∠BDA = 145°. Find the value of ∠CDB
A. 75°
B. 115°
C. 35°
D. 45°
Answer:
As ADB +
DAB = 120°
In triangle ABD as sum of all angles = 180°
ABD = 180–120 = 60°
As ABCD is cyclic, opposite angles have sum of 180°
Let BAD = x°
Hence BCD = 180–x
Let BDC = a°
Let DBC = z°;
ABC = 60 + z
Let ADB = y°
As sum of angles of triangle BDC = 180°
180–x + a + z = 180
a = x–z ...(1)
According to question,
x + y = 120 ..(2)
and
60 + z + y = 145...(3)
Subtracting (3) from (2)
We get
x + y–60–z–y = –25
x–z = 35
Equating from (1) we get
a = 35°
hence CDB = 35°
Question 3.
In the given figure, AB is one of the diameters of the circle and OC is perpendicular to it through the center O. If AC is 7√2 cm, then what is the area of the circle in cm2
A. 24.5
B. 49
C. 98
D. 154
Answer:
In triangle AOC AO = OC = radius of circle = r
Using Pythagoras theorem,
As area of a circle is given by =
Question 4.
In the given figure, AB is a diameter of the circle and points C and D are on the circumference such that ∠CAD = 30°and ∠CBA = 70° what is the measure of ACD?
A. 40°
B. 50°
C. 30°
D. 90°
Answer:
According to theorem the angle subtended by a diameter on the circumference of a circle is 90°
ACB = 90°
Angle subtended by chord AC on major segment = ABC = 70°
Angle subtended on minor segment = 180–70 = 110° = ADC
As sum of angles of triangle ADC = 180°
ACD = 180–30–110 = 40°
Question 5.
Angle in a semi circle is
A. obtuse angle
B. right angle
C. an acute angle
D. supplementary
Answer:
As a semicircle is formed by a diameter and the angle subtended on the circumference by the diameter is 90°.
Question 6.
Angle in a minor segment is
A. an acute angle
B. an obtuse angle
C. a right angle
D. a reflexive angle
Answer:
The angle formed by a chord in the major segment is acute whereas in the minor segment is obtuse and the sum of both angles is 180°.
Question 7.
In a cyclic quadrilateral ABCD, ∠A = 5x, ∠C = 4x the value of x is
A. 12°
B. 20°
C. 48°
D. 36°
Answer:
In a cyclic quadrilateral sum of opposite angles is 180°
Question 8.
Angle in a major segment is
A. an acute angle
B. an obtuse angle
C. a right triangle
D. a reflexive angle
Answer:
The angle formed by a chord in the major segment is acute whereas in the minor segment is obtuse and the sum of both angles is 180°.
Question 9.
If one angle of a cyclic quadrilateral is 70°, then the angle opposite to it is
A. 20°
B. 110°
C. 140°
D. 160°
Answer:
As the sum of opposite angles in a cyclic quadrilateral is 180°
Let opposite angle be x°