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Two Crescents (Posted on 2004-08-11) Difficulty: 2 of 5
Given a right triangle, draw the three following semicircles:
  1. The semicircle with diameter formed by one of the legs and extending away from the triangle.
  2. The semicircle with diameter formed by the other leg and extending away from the triangle.
  3. The semicircle with diameter formed by the hypotenuse and extending towards the triangle.

Prove that the area of the two crescents (shown in RED and BLUE) formed by the three semicircles equals the area of the triangle.

No Solution Yet Submitted by ThoughtProvoker    
Rating: 3.2000 (10 votes)

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Solution Cute | Comment 5 of 15 |

The area of the two crescents (A) is (red semicircle) + (blue semicircle) + triangle (hypotenuse semicircle) = A. So if I call the diameter of the red semicircle R, the diameter of the blue semicircle B, and the diameter of the hypotenuse semicircle H, here is what I get:

R^2 + B^2 = H^2 (Pythagorean theorem of the right triangle)

Red semicircle = 1/2*pi*(R/2)^2 = pi/8*R^2

Blue semicircle = pi/8*B^2

Hypotenuse semicircle = pi/8*H^2

A = pi/8*R^2 + pi/8*B^2 + triangle pi/8*H^2 = pi/8*(R^2 +B^2 H^2) + triangle = pi/8*(H^2 H^2) + triangle = triangle.

So A = triangle

  Posted by nikki on 2004-08-11 09:42:35
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