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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Circle

Olympiad workbook · IOQM · Circle

Two circles, S_1 and S_2 , of radii 6 units and 3 units...

Two circles,

S_1

and

S_2

, of radii 6 units and 3 units respectively, are tangent to each other externally. Let

A C

and

B D

be their direct common tangents with

A

and

B

on

S_1

, and

C

and

D

on

S_2

. Find the area of quadrilateral

A B D C

to the nearest integer. Two circles, S_1 and S_2 , of radii 6 units and 3 units respectively, are tangent to each other externally. Let A C and

Two circles, S_1 and S_2 , of radii 6 units and 3 units respectively, are tangent to each other externally. Let A C and

Correct answer

68

Step-by-step solution

Refer to diagram, Two circles, S_1 and S_2 , of radii 6 units and 3 units respectively, are tangent to each other externally. Let A C and

Two circles, S_1 and S_2 , of radii 6 units and 3 units respectively, are tangent to each other externally. Let A C and

\cos \theta=\frac{3}{9}=\frac{1}{3}

Let

\operatorname{ar}(\triangle A P R)=\Delta

then

\operatorname{ar}(\triangle B Q S)=\frac{\Delta}{4}

Because

\triangle A P R \sim \triangle B Q S

\begin{aligned} \Delta & =\frac{1}{2} A P \cdot P R \sin \theta \\ & =\frac{1}{2} 6 \cdot 2 \cdot \frac{2 \sqrt{2}}{3}=4 \sqrt{2} \end{aligned}

Area of

\square A B D C=2

\left[\frac{1}{2} \sqrt{72}(3+6)-\Delta+\frac{\Delta}{4}\right] =48 \sqrt{2}

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