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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.

Let ABCD be a parallelogram and a circle with centre O. Let sides AB, BC, CD and AD of the parallelogram touch the circle at E, F, G and H respectively.

Since the length of two tangents drawn from an external point to a circle are equal.
So, AE = AH    ...(i)
BE = BF    ....(ii)
CG = CF    ...(iii)
and DG = DH    ....(iv)
Adding (i), (ii), (iii) and (iv), we get
AE + BE + GC + DG = AH + BF + CF + DH
⇒ (AE + BE) + (GC + DG)
= (AH + DH) + (BF + CF)
⇒ 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.