1979 AHSME Problems/Problem 28
Problem 28
Circles with centers , and each have radius , where . The distance between each pair of centers is . If is the point of intersection of circle and circle which is outside circle , and if is the point of intersection of circle and circle which is outside circle , then length equals
Solution 1 (Coordinate Geometry)
The circles can be described in the cartesian plane as being centered at and with radius by the equations
.
Solving the first 2 equations gives which when substituted back in gives .
The larger root is the point B' described in the question. This root corresponds to .
By symmetry across the y-axis the length of the line segment is which is .
Solution 2 (Synthetic)
Suppose and intersect at . By the Pythagorean Theorem, and by a triangle, . Using Ptolemy’s Theorem on isosceles trapezoid , we get that After a little algebra, we get that as desired. Solasky (talk) 12:29, 27 May 2023 (EDT)
See Also
1979 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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