2017 AMC 12A Problems/Problem 19
Problem
A square with side length is inscribed in a right triangle with sides of length
,
, and
so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length
is inscribed in another right triangle with sides of length
,
, and
so that one side of the square lies on the hypotenuse of the triangle. What is
?
Solution 1
Analyze the first right triangle.
Note that and
are similar, so
. Th1s can be wr1tten 4s
. Solving,
.
Now we analyze the third triangle.
Similarly, and
are similar, so
, and
. Thus,
. Solving for
, we get
. Thus,
.
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 10 Problems and Solutions |
2017 AMC 12A (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 | |
All AMC 12 Problems and Solutions |
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