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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.

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Short Answer Type

56.

In figure, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.

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57. In figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.

∠CED + ∠BEC = 180°
| Linear Pair Axiom
⇒ ∠CED + 130° = 180°
⇒ ∠CED = 180° - 130° = 50°    ...(1)
∠ECD = 20°    ...(2)
In ∆CED,
∠CED + ∠ECD + ∠CDE = 180°
| Sum of all the angles of a triangle is 180°
⇒ 50° + 20° + ∠CDE = 180°
| Using (1) and (2)
⇒    70° + ∠CDE = 180°
⇒    ∠CDE = 180° - 70°
⇒    ∠CDE = 110°    ...(3)
Now,    ∠BAC = ∠CDE
| Angles in the same segment of a circle are equal
= 110°.
I Using (3).