Difference between revisions of "2023 AMC 10A Problems/Problem 17"
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Let <math>ABCD</math> be a rectangle with <math>AB = 30</math> and <math>BC = 28</math>. Point <math>P</math> and <math>Q</math> lie on <math>\overline{BC}</math> and <math>\overline{CD}</math> respectively so that all sides of <math>\triangle{ABP}, \triangle{PCQ},</math> and <math>\triangle{QDA}</math> have integer lengths. What is the perimeter of <math>\triangle{APQ}</math>? | Let <math>ABCD</math> be a rectangle with <math>AB = 30</math> and <math>BC = 28</math>. Point <math>P</math> and <math>Q</math> lie on <math>\overline{BC}</math> and <math>\overline{CD}</math> respectively so that all sides of <math>\triangle{ABP}, \triangle{PCQ},</math> and <math>\triangle{QDA}</math> have integer lengths. What is the perimeter of <math>\triangle{APQ}</math>? | ||
| − | + | <math>\textbf{(A) } 84 \qquad \textbf{(B) } 86 \qquad \textbf{(C) } 88 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 92</math> | |
| − | <math>\ | ||
==Solution== | ==Solution== | ||
Revision as of 20:43, 9 November 2023
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
[insert asy diagram]
Using knowledge of common Pythagorean triples and guess and check, we can find that 
is a 
-
-
triangle with side lengths 
-
-
, and 
and are 
-
-
triangles with side lengths 
-
- and 
-
-
, respectively.
Adding up the side lengths of gives 
~ItsMeNoobieboy
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. 
