2017 AMC 12B Problems/Problem 24
Problem
Quadrilateral has right angles at
and
, Triangle
~ Triangle
, and
. There is a point
in the interior of
such that Triangle
~ Triangle
and the area of Triangle
is
times the area of Triangle
. What is
Solution
Solution by TorrTar
Let ,
,
. Note that
. The Pythagorean theorem states that
. Since
, the ratios of side lengths must be equal. Since
,
and
. Let Point F be a point on
such that
is an altitude of triangle
. Note that
, so
and
can be calculated. Solving for these lengths gives
and
. Since
and
form altitudes of triangles
and
, respectively, the areas of these triangles can be calculated. Additionally, the area of triangle
can be calculated, as it is a right triangle. Solving for each of these yields:
(Minus yields a negative value)
Thus the answer is D: 2+sqrt(5)
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.