Difference between revisions of "2005 AMC 8 Problems/Problem 13"
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Revision as of 15:16, 24 October 2022
Problem
The area of polygon 
is 52 with 
, 
and 
. What is 
?
![[asy] pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4); draw(a--b--c--d--e--f--cycle); draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a); draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b); draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c); draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d); draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f); label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW); label("$E$", e, SW); label("$F$", f, SW); label("5", (0,6.5), W); label("8", (4,9), N); label("9", (8, 4.5), E); [/asy]](http://latex.artofproblemsolving.com/1/a/9/1a9b3504d94fa699730b91e95cc71d2a3c84db6b.png)
Solution
Notice that 
, so 
. Let 
be the intersection of the extensions of 
and 
, which makes rectangle 
. The area of the polygon is the area of 
subtracted from the area of .
![\[\text{Area} = 52 = 8 \cdot 9- EF \cdot 4\]](http://latex.artofproblemsolving.com/c/f/c/cfc8abb5efa3c6ac223b72d3961eea22b85e9e92.png)
Solving for the unknown, 
, therefore 
.
See Also
| 2005 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
