Tangent Chord

Geometry Level 5

The three colored circles above are tangent to each other, to chord A B AB , and to the large circle. The middle one has radius 5, and the other two on either side each have radius 4.

Find the length of chord A B AB .


The answer is 40.

Vijay Simha by Vijay Simha , 47, USA

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2 solutions

Ajit Athle
Jan 17, 2017

Let A B = l AB=l and R R the radius of the large circle. We know that the length of the common tangent between the larger ( r = 5 r=5 ) and the smaller ( r = 4 r=4 ) circles is 2 5 × 4 2\sqrt{5\times4} or 2 20 2\sqrt{20} . We now can write the following equations:
( R − 10 + 4 ) 2 + 4 × 20 = ( R − 4 ) 2 ( l / 2 ) 2 = R 2 − ( R − 10 ) 2 \begin{aligned} (R-10+4)^2+4\times 20&=(R-4)^2 \\ (l/2)^2&=R^2-(R-10)^2 \end{aligned}
which yield R = 25 R=25 and l = 40 l=40 .

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The flags were there in the original image. I've nothing to do with Bangladesh but the Indian flag is very dear to me.

Ajit Athle - 4 years, 4 months ago

Can you explain more on how you derived the two equations?

Christopher Boo - 4 years, 4 months ago

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Which two equations ?

You mean how to get A B 2 = 4 ( 2 5 2 − 1 5 2 ) \ AB^2 = 4(25^2 - 15^2) ?

I don't know either, but i use A B 2 × A B 2 = ( 50 − 10 ) × 10 \frac{AB}{2} \times \frac{AB}{2} = (50-10) \times 10 , which will also end up that A B = 40 AB = 40

Jason Chrysoprase - 4 years, 4 months ago

What's with the India and Bangladesh Flag ? You come from there ?

Jason Chrysoprase - 4 years, 4 months ago

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