2019 AMC 12B Problems/Problem 12
Problem
Right triangle with right angle at
is constructed outwards on the hypotenuse
of isosceles right triangle
with leg length
, as shown, so that the two triangles have equal perimeters. What is
?
Solution 1
Observe that the "equal perimeter" part implies that . A quick Pythagorean chase gives
.
Use the sine addition formula on angles
and
(which requires finding their cosines as well), and this gives the sine of
. Now, use
on angle
to get
.
Feel free to elaborate if necessary.
Solution 1.5 (Little bit of coordinate bash)
After using Pythagorean to find and
, we can instead notice that the angle between the y-coordinate and
is $45∘$ (Error compiling LaTeX. Unknown error_msg), and implies that the slope of that line is 1. If we draw a perpendicular from point
, we can then proceed to find the height and base of our triangle by coordinate-bashing, which turns out to be
and
respectively.
By double angle formula and difference of squares, it's easy to see that our answer is
~Solution by MagentaCobra
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |