Circle Geometry Question

Geometry Level 2

In the figure below, there is a semicircle with radius 2 and two identical semicircles within it, each with radius 1. The smaller semicircles are externally tangent to each other and share the edge with the larger semicircle. There is also a circle tangent to all three semicircles (internally tangent to the large semicircle, and externally tangent to the smaller semicircles).

What is the radius of the circle?

1 3 \frac{1}{3} 3 5 \frac{3}{5} 2 3 \frac{2}{3} 1 1 1 2 \frac{1}{2}

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3 solutions

We have formed a 90 degree triangle with hypotenuse (r+1) and sides 1 and 2-r (radius of large circle - radius of small circle).

We can form a Pythagorean Equation:

(r+1)² = 1 + (2-r)²

r² + 2r + 1 = 1 + 4 - 4r + r²

Cancel out the one and the r²

2r = 4 - 4r

6r = 4

r = 2 3 \frac{2}{3}

11 Helpful 2 Interesting 0 Brilliant 0 Confused

Applying pythagorean theorem on the right triangle formed (color yellow), we have

( 1 + r ) 2 = 1 2 + ( 2 − r ) 2 (1+r)^2 = 1^2 + (2-r)^2

1 + 2 r + r 2 = 1 2 + ( 4 − 4 r + r 2 ) 1+2r+r^2 = 1^2 + (4-4r+r^2)

1 + 2 r + r 2 = 1 + 4 − 4 r + r 2 1+2r+r^2 = 1 + 4-4r+r^2

2 r + 4 r = 4 2r+4r=4

6 r = 4 6r=4

r = 4 6 = r = \dfrac{4}{6}= 2 3 \boxed{\dfrac{2}{3}}

7 Helpful 1 Interesting 0 Brilliant 0 Confused

nice solution...

Ramiel To-ong - 4 years ago
Auro Light
May 29, 2017

Product of exterior part of secant with its entire length is equal to the square of tangent. So, r(2+r) = (2 - r)^2, => 2r = 4 - 4r, or r = 2/3.

4 Helpful 1 Interesting 0 Brilliant 0 Confused

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