2023 AMC 10A Problems/Problem 17
Contents
Problem
Let 
be a rectangle with 
and 
. Point 
and 
lie on 
and 
respectively so that all sides of 
and 
have integer lengths. What is the perimeter of 
?
Solution
We know that all side lengths are integers, so we can test Pythagorean triples for all triangles.
First, we focus on 
. The length of 
is 
, and the possible Pythagorean triples can be are 
, where the value of one leg is a factor of . Testing these cases, we get that only 
is a valid solution because the other triangles result in another leg that is greater than 
, the length of . Thus, we know that 
and 
.
Next, we move on to . The length of 
is , and the possible triples are 
and 
. Testing cases again, we get that is our triple. We get the value of 
, and 
.
We know that 
which is 
, and 
which is 
. 
is therefore a right triangle with side length ratios 
, and the hypotenuse is equal to 
.
has side lengths 
and 
so the perimeter is equal to 
~ Gabe Horn
See Also
| 2023 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
See Also
| 2023 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
