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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.

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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.

It is given that a circle inscribed in a ΔABC having sides.
AB = 8 cm,
BC = 10 cm
and    AC = 12 cm
Since, tangents drawn from an external points are equal.
So, AD = AF
BD = BE
and    CE = CF
Let AD = AF = x
BD = BE = y
and    CE = CF = z
AB = 8
⇒ AD + BD = 8
⇒ x + y = 8    ......(i)
∵ BC = 10
⇒    BE + CE = 10
⇒    y + z = 10    ....(ii)
∵ AC = 12
⇒ AF + CF = 12
⇒ x + z = 12    ...(iii)
Adding (i), (ii) and (iii), we get
2x + 2y + 2z = 30
⇒ 2(x + y + z) = 30
⇒    x + y + z = 15    ...(iv)
Putting the value of (i) in (iv), we get
8 + z = 15
⇒    z = 15 – 8
= 7 cm.
Putting the value of (ii) in (iv)
x + 10 = 15
⇒    x = 5 cm
Putting the value of (iii) in (iv), we get
y + 12 = 15 y = 3 cm
Hence,    x = AD = 5 cm
y = BE = 3 cm and    z = CF = 7 cm

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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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