Difference between revisions of "2012 AMC 10A Problems/Problem 15"
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{{AMC10 box|year=2012|ab=A|num-b=14|num-a=16}} | {{AMC10 box|year=2012|ab=A|num-b=14|num-a=16}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | <center><asy> | ||
+ | unitsize(2cm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(10pt)); | ||
+ | dotfactor=4; | ||
+ | |||
+ | pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); | ||
+ | pair D=(1,-2), E=(0,-2); pair F=(2,-1); pair G=(0.8,0); | ||
+ | draw(A--(2,0)); draw((0,-1)--F); draw(E--D); | ||
+ | draw(A--E); draw(B--D); draw((2,0)--F); | ||
+ | draw(A--F); draw(B--E); draw(C--G); | ||
+ | draw(H--(0,-1); draw(I--(0.5,-1); | ||
+ | |||
+ | pair[] ps={A,B,C,D,E,F,G}; | ||
+ | dot(ps); | ||
+ | |||
+ | label("$A$",A,N); | ||
+ | label("$B$",B,N); | ||
+ | label("$C$",C,W); | ||
+ | label("$D$",D,S); | ||
+ | label("$E$",E,S); | ||
+ | label("$F$",F,E); | ||
+ | label("$G$",G,N); | ||
+ | </asy></center> |
Revision as of 11:44, 15 October 2013
Contents
[hide]Problem
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of ?
Solution 1
intersects at a right angle, so . The hypotenuse of right triangle BED is .
Since AC=2BC, . is a right triangle so the area is just
Solution 2
Let be the origin. Then,
$\widebar{EB}$ (Error compiling LaTeX. Unknown error_msg) can be represented by the line Also, can be represented by the line
Subtracting the second equation from the first gives us . Thus, . Plugging this into the first equation gives us .
Since , is ,
and .
Thus, . The answer is .
See Also
2012 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); pair D=(1,-2), E=(0,-2); pair F=(2,-1); pair G=(0.8,0); draw(A--(2,0)); draw((0,-1)--F); draw(E--D); draw(A--E); draw(B--D); draw((2,0)--F); draw(A--F); draw(B--E); draw(C--G); draw(H--(0,-1); draw(I--(0.5,-1); pair[] ps={A,B,C,D,E,F,G}; dot(ps); label("$A$",A,N); label("$B$",B,N); label("$C$",C,W); label("$D$",D,S); label("$E$",E,S); label("$F$",F,E); label("$G$",G,N); (Error making remote request. Unknown error_msg)