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Difference between revisions of "1965 AHSME Problems/Problem 12"

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== Solution ==
 
== Solution ==
−
<math>\fbox{D}</math>
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 +
<asy>
 +
 
 +
draw((0,0)--(6,0));
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dot((0,0));
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label("A", (-0.75,-0.5));
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dot((6,0));
 +
label("C", (6.75,-0.5));
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label("$6$", (3,-1.75));
 +
 
 +
 
 +
draw((0,0)--(12*29/36, 12*sqrt(455)/36)--(6,0));
 +
dot((12*29/36, 12*sqrt(455)/36));
 +
label("B", (12*29/36+0.4, 12*sqrt(455)/36+0.75));
 +
label("$12$", (6*29/36-1.5, 6*sqrt(455)/36+1.5));
 +
label("$8$", (6*29/36+4.5, 6*sqrt(455)/36-0.5));
 +
 
 +
dot((4*29/36, 4*sqrt(455)/36));
 +
label("F", (4*29/36-0.75, 4*sqrt(455)/36));
 +
dot((4*29/36+4, 4*sqrt(455)/36));
 +
label("E", (4*29/36+4+0.65, 4*sqrt(455)/36-0.25));
 +
dot((4,0));
 +
label("D", (4,-0.75));
 +
 
 +
draw((0,0)--(4,0)--(4*29/36+4, 4*sqrt(455)/36)--(4*29/36, 4*sqrt(455)/36));
 +
 
 +
</asy>
 +
 
 +
As in the diagram, suppose the rhombus <math>ADEF</math> is inscribed in <math>\triangle ABC</math> with <math>D</math> on <math>\overline{AC}</math>, <math>E</math> on <math>\overline{BC}</math>, and <math>F</math> on <math>\overline{AB}</math>. Let the side length of the rhombus be <math>x</math>. Because a rhombus is a parallelogram, its opposite sides are parallel. Thus, by [[AA similarity]], <math>\triangle BFE \sim \triangle EDC</math>, and so <math>\frac{FE}{FB}=\frac{DC}{DE}</math>. Because <math>\overline{FE}</math> and <math>\overline{DE}</math> are sides of the rhombus, they both have length <math>x</math>. Furthermore, <math>FB=12-x</math>, and <math>DC=6-x</math>, because, combined with sides of the rhombus, they form sides of the triangle. Thus, by substituting into the proportion derived above, we see that:
 +
\begin{align*}
 +
\frac{FE}{FB}&=\frac{DC}{DE} \\
 +
\frac{x}{12-x}&=\frac{6-x}{x} \\
 +
x^2&=72-12x-6x+x^2 \\
 +
18x&=72 \\
 +
x&=4 \\
 +
\end{align*}
 +
 
 +
Thus, the side of the rhombus has length <math>\boxed{\text{(D) }4}</math>.
  
 
==See Also==
 
==See Also==
 
{{AHSME 40p box|year=1965|num-b=11|num-a=13}}
 
{{AHSME 40p box|year=1965|num-b=11|num-a=13}}
 +
{{MAA Notice}}
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]

Latest revision as of 16:02, 18 July 2024

Problem

A rhombus is inscribed in $\triangle ABC$ in such a way that one of its vertices is $A$ and two of its sides lie along $AB$ and $AC$. If $\overline{AC} = 6$ inches, $\overline{AB} = 12$ inches, and $\overline{BC} = 8$ inches, the side of the rhombus, in inches, is:

$\textbf{(A)}\ 2 \qquad \textbf{(B) }\ 3 \qquad \textbf{(C) }\ 3 \frac {1}{2} \qquad \textbf{(D) }\ 4 \qquad \textbf{(E) }\ 5$

Solution

[asy] draw((0,0)--(6,0)); dot((0,0)); label("A", (-0.75,-0.5)); dot((6,0)); label("C", (6.75,-0.5)); label("$6$", (3,-1.75)); draw((0,0)--(12*29/36, 12*sqrt(455)/36)--(6,0)); dot((12*29/36, 12*sqrt(455)/36)); label("B", (12*29/36+0.4, 12*sqrt(455)/36+0.75)); label("$12$", (6*29/36-1.5, 6*sqrt(455)/36+1.5)); label("$8$", (6*29/36+4.5, 6*sqrt(455)/36-0.5)); dot((4*29/36, 4*sqrt(455)/36)); label("F", (4*29/36-0.75, 4*sqrt(455)/36)); dot((4*29/36+4, 4*sqrt(455)/36)); label("E", (4*29/36+4+0.65, 4*sqrt(455)/36-0.25)); dot((4,0)); label("D", (4,-0.75)); draw((0,0)--(4,0)--(4*29/36+4, 4*sqrt(455)/36)--(4*29/36, 4*sqrt(455)/36)); [/asy]

As in the diagram, suppose the rhombus $ADEF$ is inscribed in $\triangle ABC$ with $D$ on $\overline{AC}$, $E$ on $\overline{BC}$, and $F$ on $\overline{AB}$. Let the side length of the rhombus be $x$. Because a rhombus is a parallelogram, its opposite sides are parallel. Thus, by AA similarity, $\triangle BFE \sim \triangle EDC$, and so $\frac{FE}{FB}=\frac{DC}{DE}$. Because $\overline{FE}$ and $\overline{DE}$ are sides of the rhombus, they both have length $x$. Furthermore, $FB=12-x$, and $DC=6-x$, because, combined with sides of the rhombus, they form sides of the triangle. Thus, by substituting into the proportion derived above, we see that: \begin{align*} \frac{FE}{FB}&=\frac{DC}{DE} \\ \frac{x}{12-x}&=\frac{6-x}{x} \\ x^2&=72-12x-6x+x^2 \\ 18x&=72 \\ x&=4 \\ \end{align*}

Thus, the side of the rhombus has length $\boxed{\text{(D) }4}$.

See Also

1965 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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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