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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.

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Long Answer Type

58. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.

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59. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

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60.
If the non-parallel sides of a trapezium are equal, prove that it is cyclic. Prove that an isosceles trapezium is cyclic.

Given: ABCD is a trapezium whose two non-parallel sides AD and BC are equal.
To Prove: Trapezium ABCD is cyclic.
Construction: Draw BE || AD.
Proof: ∵ AB || DE    | Given
AD || BE | By construction
∴ Quadrilateral ABCD is a parallelogram.

∴ ∠BAD = ∠BED    ...(1)
| Opp. ∠s of a || gm
and    AD = BE    ...(2)
| Opp. sides of a || gm
But AD = BC    ...(3) | Given
From (2) and (3),
BE = BC
∴ ∠BEC = ∠BCE    ...(4)
| Angles opposite to equal sides
∠BEC + ∠BED = 180°
| Linear Pair Axiom
⇒ ∠BCE + ∠BAD = 180°
| From (4) and (1)
⇒ Trapezium ABCD is cyclic.
| ∵ If a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic