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Difference between revisions of "2002 AMC 8 Problems/Problem 20"
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<math> \textbf{(A)}\ 1\frac{1}2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2\frac{1}2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 3\frac{1}2 </math>
 
<math> \textbf{(A)}\ 1\frac{1}2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2\frac{1}2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 3\frac{1}2 </math>
  
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==Solution==
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==Solution 1==
 
The shaded region is a right trapezoid. Assume WLOG that <math>YZ=8</math>. Then because the area of <math>\triangle XYZ</math> is equal to 8, the height of the triangle <math>XC=2</math>. Because the line <math>AB</math> is a midsegment, the top base of the triangle is <math>\frac12 AB = \frac14 YZ = 2</math>. Also, <math>AB</math> divides <math>XC</math> in two, so the height of the trapezoid is <math>\frac12 2 = 1</math>. The bottom base is <math>\frac12 YZ = 4</math>. The area of the shaded region is <math>\frac12 (2+4)(1) = \boxed{\text{(D)}\ 3}</math>.
 
The shaded region is a right trapezoid. Assume WLOG that <math>YZ=8</math>. Then because the area of <math>\triangle XYZ</math> is equal to 8, the height of the triangle <math>XC=2</math>. Because the line <math>AB</math> is a midsegment, the top base of the triangle is <math>\frac12 AB = \frac14 YZ = 2</math>. Also, <math>AB</math> divides <math>XC</math> in two, so the height of the trapezoid is <math>\frac12 2 = 1</math>. The bottom base is <math>\frac12 YZ = 4</math>. The area of the shaded region is <math>\frac12 (2+4)(1) = \boxed{\text{(D)}\ 3}</math>.
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==Solution 2==
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Since <math>A</math> and <math>B</math> are the midpoints of <math>XY</math> and <math>XZ</math>, respectively, <math>AY=AX=BX=BZ</math>.
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Draw segments <math>AC</math> and <math>BC</math>.
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Since <math>XY=XZ</math>, it means that <math>X</math> is on the perpendicular bisector of YZ. Then <math>YC=CZ</math>.
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<math>AB</math> is the line that connects the midpoints of two sides of a triangle together, which means that <math>AB</math> is parallel to and half in length of <math>YZ</math>. Then <math>AB=YC=CZ</math>.
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Since <math>AB</math> is parallel to <math>YZ</math>, and <math>XY</math> is the transversal, <math>\angle XAB=\angle AYC.</math> Similarly, <math>\angle XBA=\angle BZC.</math> Then, by SAS, <math>\triangle XAB=\triangle AYC=\triangle BZC</math>.
 +
Since corresponding parts of congruent triangles are congruent,<math>AC=BC=XA</math>.
 +
Using the fact that <math>AB</math> is parallel to <math>YZ</math>, <math>\angle ABC=\angle BCZ</math> and <math>\angle BAC=\angle ACY</math>. Also, <math>\angle ABC=\angle BCZ=\angle ACY</math> because <math>\triangle ABC</math> is isosceles.
 +
Now <math>\triangle XAB=\triangle AYC=\triangle BZC=\triangle ABC</math>.
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Draw an altitude through each of them such that each triangle is split into two congruent right triangles. Now there are a total of 8 congruent small triangles, each with area 1. The shaded area has three of these triangles, so it has area 3.
 +
 +
Despite this long proof you can easily see that each triangle is congruent to one another. Anyways, the area of the shaded region is <math>\boxed{\text{(D)}\ 3}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2002|num-b=19|num-a=21}}
 
{{AMC8 box|year=2002|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:13, 28 July 2013

Problem

The area of triangle $XYZ$ is 8 square inches. Points $A$ and $B$ are midpoints of congruent segments $\overline{XY}$ and $\overline{XZ}$. Altitude $\overline{XC}$ bisects $\overline{YZ}$. What is the area (in square inches) of the shaded region?

[asy] /* AMC8 2002 #20 Problem */ draw((0,0)--(10,0)--(5,4)--cycle); draw((2.5,2)--(7.5,2)); draw((5,4)--(5,0)); fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey); label(scale(0.8)*"$X$", (5,4), N); label(scale(0.8)*"$Y$", (0,0), W); label(scale(0.8)*"$Z$", (10,0), E); label(scale(0.8)*"$A$", (2.5,2.2), W); label(scale(0.8)*"$B$", (7.5,2.2), E); label(scale(0.8)*"$C$", (5,0), S); fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);[/asy]

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

Solution 1

The shaded region is a right trapezoid. Assume WLOG that $YZ=8$. Then because the area of $\triangle XYZ$ is equal to 8, the height of the triangle $XC=2$. Because the line $AB$ is a midsegment, the top base of the triangle is $\frac12 AB = \frac14 YZ = 2$. Also, $AB$ divides $XC$ in two, so the height of the trapezoid is $\frac12 2 = 1$. The bottom base is $\frac12 YZ = 4$. The area of the shaded region is $\frac12 (2+4)(1) = \boxed{\text{(D)}\ 3}$.

Solution 2

Since $A$ and $B$ are the midpoints of $XY$ and $XZ$, respectively, $AY=AX=BX=BZ$. Draw segments $AC$ and $BC$. Since $XY=XZ$, it means that $X$ is on the perpendicular bisector of YZ. Then $YC=CZ$. $AB$ is the line that connects the midpoints of two sides of a triangle together, which means that $AB$ is parallel to and half in length of $YZ$. Then $AB=YC=CZ$. Since $AB$ is parallel to $YZ$, and $XY$ is the transversal, $\angle XAB=\angle AYC.$ Similarly, $\angle XBA=\angle BZC.$ Then, by SAS, $\triangle XAB=\triangle AYC=\triangle BZC$. Since corresponding parts of congruent triangles are congruent,$AC=BC=XA$. Using the fact that $AB$ is parallel to $YZ$, $\angle ABC=\angle BCZ$ and $\angle BAC=\angle ACY$. Also, $\angle ABC=\angle BCZ=\angle ACY$ because $\triangle ABC$ is isosceles. Now $\triangle XAB=\triangle AYC=\triangle BZC=\triangle ABC$. Draw an altitude through each of them such that each triangle is split into two congruent right triangles. Now there are a total of 8 congruent small triangles, each with area 1. The shaded area has three of these triangles, so it has area 3.

Despite this long proof you can easily see that each triangle is congruent to one another. Anyways, the area of the shaded region is $\boxed{\text{(D)}\ 3}$.

See Also

2002 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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 AJHSME/AMC 8 Problems and Solutions

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