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Difference between revisions of "2002 AMC 8 Problems/Problem 20"

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==Problem==
 
==Problem==
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The area of triangle <math>XYZ</math> is 8 square inches. Points <math>A</math> and <math>B</math> are midpoints of congruent segments <math> \overline{XY} </math> and <math> \overline{XZ} </math>. Altitude <math> \overline{XC} </math> bisects <math> \overline{YZ} </math>. What is the area (in square inches) of the shaded region?
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The area of triangle <math>XYZ</math> is 8 square inches. Points <math>A</math> and <math>B</math> are midpoints of congruent segments <math>\overline{XY}</math> and <math>\overline{XZ}</math>. Altitude <math>\overline{XC}</math> bisects <math>\overline{YZ}</math>. The area (in square inches) of the shaded region is
  
 
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<asy>
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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 1==
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==Solution 3==
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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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We know the area of triangle <math>XYZ</math> is <math>8</math> square inches. The area of a triangle can also be represented as <math>\frac{bh}{2}</math> or in this problem <math>\frac{XC\cdot YZ}{2}</math>. By solving, we have <cmath>\frac{XC\cdot YZ}{2} = 8,</cmath> <cmath>XC\cdot YZ = 16.</cmath>  
  
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==Solution 2==
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With SAS congruence, triangles <math>XCY</math> and <math>XCZ</math> are congruent. Hence, triangle <math>XCY = \frac{8}{2} = 4</math>. (Let's say point <math>D</math> is the intersection between line segments <math>XC</math> and <math>AB</math>.) We can find the area of the trapezoid <math>ADCY</math> by subtracting the area of triangle <math>XAD</math> from <math>4</math>.  
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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>.
 
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Since corresponding parts of congruent triangles are congruent,<math>AC=BC=XA</math>.
 
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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.
 
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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.
 
  
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Basically the proof is to show <math>\triangle XAB=\triangle AYC=\triangle BZC=\triangle ABC</math>. If you just look at the diagram you can easily see that the triangles are congruent and you would solve this a lot faster. Anyways, the area of the shaded region is <math>\boxed{\text{(D)}\ 3}</math>.
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We find the area of triangle <math>XAD</math> by the <math>\frac{bh}{2}</math> formula- <math>\frac{XD\cdot AD}{2} = \frac{\frac{XC}{2}\cdot AD}{2}</math>. <math>AD</math> is <math>\frac{1}{4}</math> of <math>YZ</math> from solution 1. The area of <math>XAD</math> is <cmath>\frac{\frac{XC}{2}\cdot \frac{YZ}{4}}{2} = \frac{16}{16} = 1</cmath>.
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Therefore, the area of the shaded area- trapezoid <math>ADCY</math> has area <math>4-1 = \boxed{\text{(D)}\ 3}</math>.  
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- sarah07
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==Video Solution==
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https://www.youtube.com/watch?v=zwy5U5IQi88  ~David
  
 
==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}}

Latest revision as of 10:00, 14 June 2024

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 $\overline{YZ}$. The area (in square inches) of the shaded region is

[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 3

We know the area of triangle $XYZ$ is $8$ square inches. The area of a triangle can also be represented as $\frac{bh}{2}$ or in this problem $\frac{XC\cdot YZ}{2}$. By solving, we have \[\frac{XC\cdot YZ}{2} = 8,\] \[XC\cdot YZ = 16.\]

With SAS congruence, triangles $XCY$ and $XCZ$ are congruent. Hence, triangle $XCY = \frac{8}{2} = 4$. (Let's say point $D$ is the intersection between line segments $XC$ and $AB$.) We can find the area of the trapezoid $ADCY$ by subtracting the area of triangle $XAD$ from $4$.

We find the area of triangle $XAD$ by the $\frac{bh}{2}$ formula- $\frac{XD\cdot AD}{2} = \frac{\frac{XC}{2}\cdot AD}{2}$. $AD$ is $\frac{1}{4}$ of $YZ$ from solution 1. The area of $XAD$ is \[\frac{\frac{XC}{2}\cdot \frac{YZ}{4}}{2} = \frac{16}{16} = 1\].

Therefore, the area of the shaded area- trapezoid $ADCY$ has area $4-1 = \boxed{\text{(D)}\ 3}$.


- sarah07

Video Solution

https://www.youtube.com/watch?v=zwy5U5IQi88 ~David

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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