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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?

Since the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the point of contact at the centre,
i.e., ∠POQ + ∠PTQ=180°
But ∠PTQ = 70° (given)
∴ ∠POQ = 108° – ∠PTQ
= 180° – 70° = 110°

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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.

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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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