1980 AHSME Problems/Problem 19
Problem
Let and be three parallel chords of a circle on the same side of the center. The distance between and is the same as the distance between and . The lengths of the chords are , and . The radius of the circle is
Solution
Let the center of the circle be on the origin with equation . As the chords are bisected by the x-axis their y-coordinates are respectively. Let the chord of length have x-coordinate . Let be the common distance between chords. Thus, the coordinates of the top of the chords will be for the chords of length , and respectively. As these points fall of the circle, we get three equations: Subtracting the first equation from the second we get: Similarly, by subtracting the first equation from the third we get: Subtracting these two equations gives us . Expanding the second equation now gives us Subtracting the first equation from this yields: Combining this with we get Plugging this into the first equation finally us
See also
1980 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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