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Difference between revisions of "2023 AMC 10A Problems/Problem 17"

(→‎Solution)
(→‎Solution: Added a diagram)
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==Solution==
 
==Solution==
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<asy>
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/* ~ItsMeNoobieboy */
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size(200);
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pair A, B, C, D, P, Q;
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A = (0,28/30);
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B = (1,28/30);
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C = (1,0);
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D = (0,0);
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P = (1,12/30);
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Q = (21/30,0);
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draw(A--B--C--D--cycle);
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draw(A--P--Q--cycle);
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dot("$A$",A,NW,linewidth(4));
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dot("$B$",B,NE,linewidth(4));
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dot("$C$",C,SE,linewidth(4));
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dot("$D$",D,SW,linewidth(4));
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dot("$P$",P,E,linewidth(4));
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dot("$Q$",Q,S,linewidth(4));
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label("$30$",midpoint(A--B),N);
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label("$16$",midpoint(B--P),E);
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label("$34$",midpoint(A--P),NE, red);
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label("$28$",midpoint(A--D),W);
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label("$21$",midpoint(D--Q),S);
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label("$35$",midpoint(A--Q),SW, red);
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label("$9$",midpoint(Q--C),S);
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label("$12$",midpoint(C--P),E);
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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 10:27, 10 November 2023

Problem

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$?

$\textbf{(A) } 84 \qquad \textbf{(B) } 86 \qquad \textbf{(C) } 88 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 92$

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.

First, we focus on $\triangle{ABP}$. The length of $AB$ is $30$, and the possible Pythagorean triples $\triangle{ABP}$ can be are $(3, 4, 5), (5, 12, 13), (8, 15, 17),$ where the value of one leg is a factor of $30$. Testing these cases, we get that only $(8, 15, 17)$ is a valid solution because the other triangles result in another leg that is greater than $28$, the length of $\overline{BC}$. Thus, we know that $BP = 16$ and $AP = 34$.

Next, we move on to $\triangle{QDA}$. The length of $AD$ is $28$, and the possible triples are $(3, 4, 5)$ and $(7, 24, 25)$. Testing cases again, we get that $(3, 4, 5)$ is our triple. We get the value of $DQ = 21$, and $AQ = 35$.

We know that $CQ = CD - DQ$ which is $9$, and $CP = BC - BP$ which is $12$. $\triangle{CPQ}$ is therefore a right triangle with side length ratios ${3, 4, 5}$, and the hypotenuse is equal to $15$. $\triangle{APQ}$ has side lengths $34, 35,$ and $15,$ so the perimeter is equal to $34 + 35 + 15 = \boxed{\textbf{(A) } 84}.$

~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

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