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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.

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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.

DS = DR (Tangents from external point D.)
But    DS = 5 cm.
∴ DR = 5 cm.
It is given that,
AD = 23 cm.
⇒ AR + DR = 23 cm.
⇒    AR + 5 = 23
⇒    AR = 18 cm
Now, AR = AQ (Tangents from external point A)
But    AR = 18 cm
∴ AQ =18 cm.
It is given that,
AB = 29 cm.
⇒ AQ + BQ =29
⇒ 18 + BQ = 29
⇒ BQ = 11 cm.
It is given that ∠B = 90°.
Also, OQ = QP, therefore OPBQ is a square.
So,    OQ = OP = 11 cm.
Hence,    r = 11 cm.

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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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