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# Radius of a Circle – SMO 2013 – Problem 25

Try this beautiful problem from Geometry based on the radius and tangent of a circle. You may use sequential hints to solve the problem.

Try this beautiful problem from Geometry based on the radius and tangent of a circle.

## SMO 2013 – Geometry (Problem 25)

As shown in the figure below ,circles $C_1$and$C_2$ of radius 360 are tangent to each other , and both tangent to the straight line l.if the circle$C_3$ is tangent to $C_1$ ,$C_2$ and l ,and circle$C_4$is tangent to$C_1$,$C_3$ and l ,find the radius of$C_4$

• 30
• 35
• 40

### Key Concepts

Geometry

Pythagoras theorm

Distance Formula

But try the problem first…

Source

Pre College Mathematics

## Try with Hints

First hint

Let R be the radius of $C_3$

$C_2E$ =360-R

$C_3E=360$

$C_2C_3$=360+R

Using pythagoras theorm ….

$(360-R)^2+360^2=(360+R)^2$

i.e R=90

Can you now finish the problem ……….

Second Hint

Let the radius of$C_4$ be r

then use the distacce formula and tangent property……..

can you finish the problem……..

Final Step

Let r be the radius of $C_4$ (small triangle).

LO+OC=360

$\sqrt{(360+p)^2-(360-p)^2}+\sqrt{(90+r)^2-(90-r)^2}=360$

i.e r=40.

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