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##### Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2018. Subjective Problem no. 2.

Plane Geometry

5.5 out of 10

##### Suggested Book

‘Challenge and Thrill of Pre-College Mathematics’ by V,Krishnamurthy, C.R.Pranesachar, ect.

Do you really need a hint? Try it first!

$$PQ$$ and $$RS$$ are two chords of the circle $$C$$ , intersecting at the point $$O$$. See figure: click here.

Given $$PO=3$$ cm

$$SO=4$$ cm

$$[\triangle POR]= 7 cm^2$$.

From the triangles $$POS$$ and $$QOS$$ we have,

$$\angle POR=\angle SOQ$$ [Opposite angles]

$$\angle SRP=\angle SQO$$ [Angle on the same semi-circle $$STP$$]

$$\angle QSO= \angle OPR$$ [Angle on the same semi-circle $$ST’P$$]

Therefore the $$\triangle POR$$ and $$\triangle SOQ$$ are similar triangles .

$$\frac{[\triangle POR]}{OP^2}=\frac{[\triangle SOQ]}{SO^2}.$$

$$\Rightarrow [\triangle SOQ]=\frac{SO^2}{PO^2}\cdot [\triangle POR]$$=$$\frac{4^2}{3^2}\cdot 7=12\frac{4}{9}$$.(Ans.)

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