Difference between revisions of "2002 AMC 8 Problems/Problem 20"
Mrdavid445 (talk | contribs) (Created page with "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> \ov...") |
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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? | 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? | ||
| − | + | <asy> | |
/* AMC8 2002 #20 Problem */ | /* AMC8 2002 #20 Problem */ | ||
draw((0,0)--(10,0)--(5,4)--cycle); | draw((0,0)--(10,0)--(5,4)--cycle); | ||
| Line 7: | Line 7: | ||
draw((5,4)--(5,0)); | draw((5,4)--(5,0)); | ||
fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey); | fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey); | ||
| − | label(scale(0.8)*" | + | label(scale(0.8)*"$X$", (5,4), N); |
| − | label(scale(0.8)*" | + | label(scale(0.8)*"$Y$", (0,0), W); |
| − | label(scale(0.8)*" | + | label(scale(0.8)*"$Z$", (10,0), E); |
| − | label(scale(0.8)*" | + | label(scale(0.8)*"$A$", (2.5,2.2), W); |
| − | label(scale(0.8)*" | + | label(scale(0.8)*"$B$", (7.5,2.2), E); |
| − | label(scale(0.8)*" | + | label(scale(0.8)*"$C$", (5,0), S); |
| − | fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white); | + | fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);</asy> |
<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> | ||
Revision as of 20:40, 28 July 2011
The area of triangle 
is 8 square inches. Points 
and 
are midpoints of congruent segments 
and 
. Altitude 
bisects 
. 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]](http://latex.artofproblemsolving.com/3/a/8/3a879a53da609c8f6948eef78732e18f14a90fe4.png)
