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Difference between revisions of "1960 AHSME Problems/Problem 37"

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<cmath>n = \frac{b(h-x)}{h}</cmath>
 
<cmath>n = \frac{b(h-x)}{h}</cmath>
 
Thus, the area of the rectangle is <math>\frac{bx}{h}(h-x)</math>, which is answer choice <math>\boxed{\textbf{(A)}}</math>.
 
Thus, the area of the rectangle is <math>\frac{bx}{h}(h-x)</math>, which is answer choice <math>\boxed{\textbf{(A)}}</math>.
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==Video Solution==
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https://youtu.be/ZdM2ou5Gsuw?t=172
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~MathProblemSolvingSkills.com
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==See Also==
 
==See Also==
 
{{AHSME 40p box|year=1960|num-b=36|num-a=38}}
 
{{AHSME 40p box|year=1960|num-b=36|num-a=38}}
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[[Category:Intermediate Geometry Problems]]

Latest revision as of 21:26, 28 December 2023

Problem

The base of a triangle is of length $b$, and the altitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:

$\textbf{(A)}\ \frac{bx}{h}(h-x)\qquad \textbf{(B)}\ \frac{hx}{b}(b-x)\qquad \textbf{(C)}\ \frac{bx}{h}(h-2x)\qquad \textbf{(D)}\ x(b-x)\qquad \textbf{(E)}\ x(h-x)$

Solution

Let $AB=b$, $DE=h$, and $WX = YZ = x$. [asy] pair A=(0,0),B=(56,0),C=(20,48),D=(20,0),W=(10,0),X=(10,24),Y=(38,24),Z=(38,0); draw(A--B--C--A); draw((10,0)--(10,24)--(38,24)--(38,0)); draw(C--D); dot(A); dot(B); dot(C); dot(D); dot(W); dot(X); dot(Y); dot(Z); dot((20,24)); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,S); label("$W$",W,S); label("$X$",X,NW); label("$Y$",Y,NE); label("$Z$",Z,S); label("$N$",(20,24),NW); [/asy] Since $CD$ is perpendicular to $AB$, $ND = WX$. That means $CN = h-x$. The sides of the rectangle are parallel, so $XY \parallel WZ$. That means by AA Similarity, $\triangle CXY \sim \triangle CAB$. Letting $n$ be the length of the base of the rectangle, that means \[\frac{h-x}{n} = \frac{h}{b}\] \[n = \frac{b(h-x)}{h}\] Thus, the area of the rectangle is $\frac{bx}{h}(h-x)$, which is answer choice $\boxed{\textbf{(A)}}$.


Video Solution

https://youtu.be/ZdM2ou5Gsuw?t=172

~MathProblemSolvingSkills.com


See Also

1960 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 36 		Followed by
Problem 38
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions
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