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Circles

Short Answer Type

51. Two circles of radii 10 cm and 8 cm intersect and the length of the common chord is 12 cm. Find the distance between their centres.

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52. A chord 12 cm long is 8 cm away from the centre of the circle. What is the length of a chord which is 6 cm away from the centre?

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53.

In figure A,B and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

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54. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

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Long Answer Type

55. In figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.

Take a point S in the major arc. Join PS and RS.

∵ PQRS is a cyclic quadrilateral.
∴ ∠PQR + ∠PSR = 180°
| The sum of either pair of opposite angles of a cyclic quadrilateral is 180°
⇒ 100° + ∠PSR = 180°
⇒    ∠PSR = 180° - 100°
⇒    ∠PSR = 80°    ...(1)
Now, ∠POR = 2∠PSR
| The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle
= 2 × 80° = 160°    ...(2)
I Using (1)
In ∆OPR,
∵ OP = OR    | Radii of a circle
∴ ∠OPR = ∠ORP    ...(3)
| Angles opposite to equal sides of a triangle are equal
In ∆OPR,
∠OPR + ∠ORP + ∠POR = 180°
| Sum of all the angles of a triangle is 180°
⇒ ∠OPR + ∠OPR + 160° = 180°
| Using (2) and (1)
⇒    2∠OPR + 160° = 180°
⇒ 2∠OPR = 180° - 160° = 20°
⇒ ∠OPR = 10°.