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Circles

Short Answer Type

231.

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

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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

In ΔMAO and ΔCAO,
AM = AC
(Tangents drawn from external point to a
circle are equal) AO = AO    (Common)
and    OM = OC (radii of circle)
Therefore, by SSS congruent condition
ΔAMO ≅ ΔAOC
⇒ ∠OAM = ∠OAC
⇒     ∠MAB = 2 ∠OAC
or    ∠MAB = 2 ∠OAB
Similarly, we can prove
∠NBA = 2 ∠OBC
or    ∠NBA = 2 ∠OBA
Now, ∠MAB + ∠NBA = 180°
(Consecutive interior angles of the same side of transversal)
⇒ 2∠OAB + 2∠OBA = 180°
⇒ (∠OAB + ∠OBA) = 90°
Now, in Δ AOB
(∠OAB + ∠OBA + ∠AOB = 180°
(angle sum property of triangle)
Δ    90° + ∠AOB = 180°
⇒    ∠AOB = 90.

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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.