A Curious Crescent

Strategies and Hints

  1. The angle must measure 90º; that is, any triangle with a base that is the diameter of a circle is a right triangle.

  1. The right triangle needed for the solution is . If the radius is x units long, the altitude has a length of x -5.

  2. The distance from O to the left edge of the crescent has a length of x - 9. In this figure, this segment is OY.

  3. Using the two similar right triangles OYX and OXZ, this proportion can be written:



    Using x for ZO, .

Solution

Solving the proportion for x gives a radius of 25 units for the larger circle. The two diameters are 50 and 41 units.