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Problem
Triangle has a right angle at
and
. Triangle
has a right angle at
and
. Points
and
are on opposite sides of
. The line through
parallel to
meets
extended at
. If
where
and
are relatively prime positive integers, then
Solution 1
Solution by e_power_pi_times_i
Let be the point such that
and
are parallel to
and
, respectively, and let
and
. Then,
. So,
. Simplifying
, and
. Therefore
, and
. Checking,
is the answer, so
. The answer is
.
Solution 2
Solution by Arjun Vikram
Extend lines and
to meet at a new point
. Now, we see that
. Using this relationship, we can see that
, (so
), and the ratio of similarity between
and
is
. This ratio gives us that
. By the Pythagorean Theorem,
. Thus,
, and the answer is
.
Solution 3 (Trig)
We have and
Now we are trying to find
Now we use the
angle sum identity, which states
Using this identity yields
See also
1991 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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