1953 AHSME Problems/Problem 48
Problem
If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is:
Solution
Let be the length of the smaller base of isosceles trapezoid , and be the length of the larger base of the trapezoid. The ratio of the smaller base to the larger base is . Let point be the foot of the altitude from to .
Since the larger base of the isosceles trapezoid equals a diagonal, . Since the smaller base equals the altitude, . Since the trapezoid is isosceles, , so . Using the Pythagorean Theorem on right triangle , we have Multiplying both sides by gives Expanding the squared binomial and rearranging gives This can be factored into . Since a must be positive, , so . The ratio of the smaller base to the larger base is .
See Also
1953 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 47 |
Followed by Problem 49 | |
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