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261. A circle is touching the side BC of ΔABC at P and touching AB and AC produced at Q and R respectively. Prove that

$AQ space equals space 1 half space left parenthesis Perimeter space of space increment space ABC right parenthesis$

Since, the length of tangents drawn from an external point to a circle are equal.
So, BQ = BP
CP = CR and AQ = AR
Now, perimeter of ΔABC,
= AB + BC + AC
= AB + BP + PC + AC
= AB + BQ + CR + AC

[∴ PB = BQ, PC = CR] = AQ + AR
= 2 AQ [∴ AQ = AR]

$rightwards double arrow space space space thin space AQ space equals space 1 half space left parenthesis space perimeter space of space increment space ABC right parenthesis$

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262. In the given figure, the incircle of ΔABC, touches the sides BC, CA and AB at D, E and F respectively. Show that AF + BD + CE = AE + BF + CD

equals space 1 half left parenthesis perimeter space of space increment space ABC right parenthesis

334 Views

Long Answer Type

263. ABCD is a quadrilateral such that ∠D = 90°. A cirlce C (O, r) touches the sides AB, BC, CD and DA at P, Q, R and S respectively. If BC = 38 cm, CD = 25 cm and BP = 27 cm, Find r.

358 Views

Short Answer Type

264. In figure. XP and XQ are tangents from X to the circle with ccentre O. R is a point on the circle. Prove that XA + AR = XB + BR?

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265. In figure a circle touches the side BC of ΔABC at P and touches AB and AC produced at Q and R respectively. If AQ = 5 cm, find the perimeter of ΔABC.

265 Views

Long Answer Type

266. In the given figure, ABC is a right-angled triangle with AB = 6 cm and AC = 8 cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of r, the radius of the inscribed circle.

421 Views

Short Answer Type

267.

From a point P outside a circle, with centre O tangents PA and PB are drawn as shown in the figure, prove that

(i) ⇒AOP = ∠BOP
(ii) OP is the perpendicular bisector of AB.

1204 Views

Long Answer Type

268. The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle. BD is a tangent to the smaller circle touching it at D. Find the length AD.

3545 Views

Short Answer Type

269. In two concentric circles, prove that all chords of the outer circle which touches the inner circle are of equal length.

3238 Views

270. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre of the circle.

625 Views