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Circles

Short Answer Type

251. Two tangents TP and TQ are drawn from an external point T to a circle with centre O, as shown in figure. If they are inclined to each other at angle of 70°, then what is the value of ∠POQ?

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252. In the figure given below, PA and PB are tangents to the circle drawn from an external point P. CD is a third tangent touching the circle at Q. If PB = 10 cm and CQ = 2 cm, what is the length of PC?

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253. The length of tangent from a point A at a distance of 5 cm from the centre of the circle is 4 cm. What will be the radius of the circle?

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254. What is the distance between two parallel tangents of a circle of the radius 4 cm?

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255. Two tangents TP and TQ are drawn from an external point T to a circle with centre O, as shown in Fig. If they are inclined to each other at an angle of 100°, then what is the value of ∠POQ?

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256. In figure 2, if ∠ATO = 40°, find ∠AOB.

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257. Two tangents PA and PB are drawn to a cirlce with centre O from an external point P. Prove that ∠APB = 2∠OAB.

Given : A circle with centre O, an external point P and two tangents PA and PB to the circle, where A and B are points of contact (Fig. 10.36A).
To Prove : ∠APB 2∠OAB
Proof : Let ∠APB = 9 ∴ PA = PB
[∴ Tangents from an ext. pt. are equal]

$rightwards double arrow$ $increment$ APB is ab isosceles triangle

$rightwards double arrow space space space space space space space space space space angle PAB space equals space angle PBA space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half left parenthesis 180 degree minus angle APB right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half left parenthesis 180 degree minus straight theta right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space equals space 90 degree minus 1 half straight theta$
$But space space space space space space space space space space angle OAP space equals space 90 degree space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space Radius space is space perpendicular space to space the space tangent right square bracket therefore space space space space space space space angle OAB space equals space angle OAP space minus space angle PAB space space space space space space space space space space space space space space space space space space space space space equals space 90 degree minus open parentheses 90 degree minus 1 half straight theta close parentheses space space space space space space space space space space space space space space space space space space space space space space equals space 1 half straight theta equals 1 half angle APB space space space space space space space space space space space space space space space space space space space space space$

Hence, ∠APB = 2 ∠OAB. Proved.

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258. O is the centre of two concentric circles of radii 4 cm and 6 cm. PT and PS are tangents to the larger circle and smaller circle respectively from an external point P.PT = 10 cm, find the length of PS.

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259. A circle is inscribed in a ΔABC, having sides 8 cm, 10 cm, and 12 cm, as shown in Fig. Find AD, BE and CF.

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260. In figure, a circle touches all the four sides of a quadrilateral ABCD_/ whose sides AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

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