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

Revision as of 19:17, 23 December 2012

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

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 $\frac12 (2+4)(1) = \boxed{\text{(D)}\ 3}$.

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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