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Circles

Short Answer Type

341.

In Figure 1, common tangents AB and CD to the two circles with centres O₁ and O₂ intersect at E. Prove that AB = CD.

342.

The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC.

Long Answer Type

343.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

344.
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Let ABCD be a quadrilateral circumscribing a circle centered at O such that it touches the circle at point P, Q, R, S.

Let us join the vertices of the quadrilateral ABCD to the center of the circle.

$Consider ∆ OAP and ∆ OAS,$

AP = AS (Tangents from the same point)

OP = OS (Radii of the same circle)

OA = OA ( common side )

$∆ OAP ≅ ∆ OAS (SSS congruence criterion) Therefore, A \leftrightarrow A, P \leftrightarrow S, O \leftrightarrow O And thus, ∠ POA = ∠ AOS ∠ 1 = ∠ 8 Similarly, ∠ 2 = ∠ 3 ∠ 4 = ∠ 5 ∠ 6 = ∠ 7 ∠ 1 + ∠ 2 + ∠ 3 + ∠ 4 + ∠ 5 + ∠ 6 + ∠ 7 + ∠ 8 = 360 ° (∠ 1 + ∠ 8) + (∠ 2 + ∠ 3) + (∠ 4 + ∠ 5) + (∠ 6 + ∠ 7) = 360 ° 2 ∠ 1 + 2 ∠ 2 + 2 ∠ 5 + 2 ∠ 6 = 360 ° 2 (∠ 1 + ∠ 2) + 2 (∠ 5 + ∠ 6) = 360 ° (∠ 1 + ∠ 2) + (∠ 5 + ∠ 6) = 180 ° ∠ AOB + ∠ COD = 180 ° Similarly, we can prove that ∠ BOC + ∠ DOA = 180 °$

Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Short Answer Type

345.

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP.

346.

A circle touches all the four sides of a quadrilateral ABCD. Prove that

AB + CD = BC + DA

347.

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

Long Answer Type

348.

In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangents AB with point of contact C, is intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°.

349.

Prove that the lengths of two tangents drawn from an external point to a circle are equal.

Multiple Choice Questions

350.

In Fig., the sides AB, BC and CA of a triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then the length of BC (in cm) is