2019 AMC 10B Problems/Problem 16
Problem
In with a right angle at
, point
lies in the interior of
and point
lies in the interior of
so that
and the ratio
. What is the ratio
Solution
Without loss of generality, let and
. Let
and
. As
and
are isosceles,
and
. Then
, so
is a 3-4-5 triangle with
.
Then , and
is a 1-2-
triangle.
On isosceles triangles and
, drop altitudes from
and
onto
; denote the feet of these altitudes by
and
respectively. Then
by AAA similarity, so we get that
, and
. Similarly we get
, and
.
-scrabbler94
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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 | |
All AMC 12 Problems and Solutions |
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