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231.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that
AB + CD = AD + BC.

Since, tangents drawn from an exterior point to a circle are equal in length.

∴ AP = AS    ...(i)
BP = BQ    ...(ii)
CR = CQ    ...(iii)
and DR = DS    ...(iv)
Adding (i), (ii), (iii) and (iv), we get
AP + BP + CR + DR = AS + BQ + CQ + DS
⇒ (AP + BP) + (CR + DR)
= (AS + DS) + (BQ + CQ)
⇒ AB + CD = AD + BC
Hence, AB + CD = BC + DA.

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232. In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.

Fig. 10.13

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233.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

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234. Prove that the parallelogram circumscribing a circle is a rhombus.

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Long Answer Type

235.

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Fig, 10.14

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236.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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Short Answer Type

237. From a point P, the length of the tangent to a circle is 15 cm and distance of P from the centre of the circle is 17 cm. Then what is the radius of the circle?

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238. In Fig. 10.18, PA is a tangent from an external point P to a circle with centre O. If ∠POB = 115°, then find ∠APO.

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239. In Fig. 10.19, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 7 cm, then find the length of BR.