2008 iTest Problems/Problem 89
Revision as of 18:53, 23 November 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 89 -- perpendicular circles in spheres)
Problem
Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points, and , such that . If the radii of the two circles are and , find , where is the radius of the sphere.
Solution
Let be the center of the sphere, be the center of the circle radius , be the center of the circle radius , and be the midpoint of . Since and , by the Pythagorean Theorem, and .
Additionally, by symmetry, the plane containing must also contain . Since the two planes are perpendicular, . Because and , is a rectangle, so .
Thus, by the Pythagorean Theorem the radius of the circle is , so .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 88 |
Followed by: Problem 90 | |
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