Difference between revisions of "2023 AMC 10A Problems/Problem 17"
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==Solution== | ==Solution== | ||
| + | <asy> | ||
| + | /* ~ItsMeNoobieboy */ | ||
| + | size(200); | ||
| + | pair A, B, C, D, P, Q; | ||
| + | A = (0,28/30); | ||
| + | B = (1,28/30); | ||
| + | C = (1,0); | ||
| + | D = (0,0); | ||
| + | P = (1,12/30); | ||
| + | Q = (21/30,0); | ||
| + | draw(A--B--C--D--cycle); | ||
| + | draw(A--P--Q--cycle); | ||
| + | dot("$A$",A,NW,linewidth(4)); | ||
| + | dot("$B$",B,NE,linewidth(4)); | ||
| + | dot("$C$",C,SE,linewidth(4)); | ||
| + | dot("$D$",D,SW,linewidth(4)); | ||
| + | dot("$P$",P,E,linewidth(4)); | ||
| + | dot("$Q$",Q,S,linewidth(4)); | ||
| + | label("$30$",midpoint(A--B),N); | ||
| + | label("$16$",midpoint(B--P),E); | ||
| + | label("$34$",midpoint(A--P),NE, red); | ||
| + | label("$28$",midpoint(A--D),W); | ||
| + | label("$21$",midpoint(D--Q),S); | ||
| + | label("$35$",midpoint(A--Q),SW, red); | ||
| + | label("$9$",midpoint(Q--C),S); | ||
| + | label("$12$",midpoint(C--P),E); | ||
| + | label("$15$",midpoint(Q--P),SE, red); | ||
| + | </asy> | ||
| + | |||
We know that all side lengths are integers, so we can test Pythagorean triples for all triangles. | We know that all side lengths are integers, so we can test Pythagorean triples for all triangles. | ||
Revision as of 09:27, 10 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
![[asy] /* ~ItsMeNoobieboy */ size(200); pair A, B, C, D, P, Q; A = (0,28/30); B = (1,28/30); C = (1,0); D = (0,0); P = (1,12/30); Q = (21/30,0); draw(A--B--C--D--cycle); draw(A--P--Q--cycle); dot("$A$",A,NW,linewidth(4)); dot("$B$",B,NE,linewidth(4)); dot("$C$",C,SE,linewidth(4)); dot("$D$",D,SW,linewidth(4)); dot("$P$",P,E,linewidth(4)); dot("$Q$",Q,S,linewidth(4)); label("$30$",midpoint(A--B),N); label("$16$",midpoint(B--P),E); label("$34$",midpoint(A--P),NE, red); label("$28$",midpoint(A--D),W); label("$21$",midpoint(D--Q),S); label("$35$",midpoint(A--Q),SW, red); label("$9$",midpoint(Q--C),S); label("$12$",midpoint(C--P),E); label("$15$",midpoint(Q--P),SE, red); [/asy]](http://latex.artofproblemsolving.com/4/1/f/41fc6aee278a2183fa354f7d7f71a911b18f7b29.png)
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 ~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. 
