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Circles

Short Answer Type

271.

The incircle of ΔABC touches the sides BC, CA and BA at D, E and F respectively. If
AB = AC, prove that BD = CD.

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272. In Figure 7, TP and TQ are tangents from T to the circle with centre O and R is any point on the circle. If AB is a tangent to the circle is R, prove that TA + AR = TB + BR.

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273. Prove that the lengths of tangent s drawn from an external point to a circle are equal.

Given : A circle C (O, r), in which AP and AQ are two tangents drawn from an external point A.
To prove :    AP = AQ
Const. : Join O P, OQ and OA.
Proof : Since the tangent at any point of a circle is perpendicular to the radius through
the point of contact.
∴ ∠OPA = ∠OQA = 90°    ...(i)
Now, in right triangles OP A and OQA, We have
OP = OQ    (radii of circle)
OA = OA    (common)
and ∠OPA = ∠OQA     from (i)
Therefore,
ΔOPA = ΔOQA    (By RHS)
This gives AP = AQ    (Hence proved)

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274.

A quadrilateral ABCD is drawn to circumscribe a circle (Fig. 10.62). Prove that AB + CD = AD + BC.

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275. In Figure, a circle is inscribed in a quadrilateral ABCD in which DB = 90°. If AD = 23 cm. and DS = 5 cm. find the radius (r) of the circle.

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276. ABC is a isosceles triangle in which AB = AC circumscribed about a circle. Show that BC is bisected at the point of contact.

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277. In Fig. 10.65A, quadrilateral ABCD is circumscribing a circle. Find the perimeter of the quadrilateral ABCD.

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278. A circle is inscribed in a ΔABC, touching AB, BC and AC at P,Q and R respectively, as shown in Fig. 10.65 B. If AB = 10 cm, AR = 7 cm and RC = 5 cm, then find the length of BC.

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279. Using the above, do the following : In Fig., from a point P, tangents PT and PS are drawn to a circle with centre O. At a point C on the circle, another tangent is drawn to the circle to intersect PT in A and PS in B. If PT = 12 cm and BC = 5 cm, find the length of PB.

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280. If AB, AC, PQ are tangents in figure, and AB = 7 cm, and XQ = 2 cm, find the length of AQ.

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