1985 AIME Problems/Problem 9
Problem
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of , , and radians, respectively, where . If , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
Solution 1
All chords of a given length in a given circle subtend the same arc and therefore the same central angle. Thus, by the given, we can re-arrange our chords into a triangle with the circle as its circumcircle.
This triangle has semiperimeter so by Heron's formula it has area . The area of a given triangle with sides of length and circumradius of length is also given by the formula , so and .
Now, consider the triangle formed by two radii and the chord of length 2. This isosceles triangle has vertex angle , so by the Law of Cosines,
and the answer is .
Solution 2 (Law of Cosines)
It’s easy to see in triangle which lengths 2, 3, and 4, that the angle opposite the side 2 is , and using the Law of Cosines, we get: Which, rearranges to: And, that gets us: Using , we get that: Which gives an answer of
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Solution 3 (trig)
Using the first diagram above, by the Pythagorean trig identities, so by the composite sine identity multiply both sides by , then subtract from both sides squaring both sides, we get plugging this back in, so and the answer is
See also
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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