Difference between revisions of "2013 AMC 8 Problems/Problem 24"
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==Problem== | ==Problem== | ||
| + | Squares <math>ABCD</math>, <math>EFGH</math>, and <math>GHIJ</math> are equal in area. Points <math>C</math> and <math>D</math> are the midpoints of sides <math>IH</math> and <math>HE</math>, respectively. What is the ratio of the area of the shaded pentagon <math>AJICB</math> to the sum of the areas of the three squares? | ||
| − | + | <math> \textbf{(A)}\hspace{.05in}\frac{1}{4}\qquad\textbf{(B)}\hspace{.05in}\frac{7}{24}\qquad\textbf{(C)}\hspace{.05in}\frac{1}{3}\qquad\textbf{(D)}\hspace{.05in}\frac{3}{8}\qquad\textbf{(E)}\hspace{.05in}\frac{5}{12}</math> | |
| − | == | + | <asy> |
| − | + | pair A,B,C,D,E,F,G,H,I,J; | |
| − | + | ||
| + | A = (0.5,2); | ||
| + | B = (1.5,2); | ||
| + | C = (1.5,1); | ||
| + | D = (0.5,1); | ||
| + | E = (0,1); | ||
| + | F = (0,0); | ||
| + | G = (1,0); | ||
| + | H = (1,1); | ||
| + | I = (2,1); | ||
| + | J = (2,0); | ||
| + | draw(A--B); | ||
| + | draw(C--B); | ||
| + | draw(D--A); | ||
| + | draw(F--E); | ||
| + | draw(I--J); | ||
| + | draw(J--F); | ||
| + | draw(G--H); | ||
| + | draw(A--J); | ||
| + | filldraw(A--B--C--I--J--cycle,grey); | ||
| + | draw(E--I); | ||
| + | dot("$A$", A, NW); | ||
| + | dot("$B$", B, NE); | ||
| + | dot("$C$", C, NE); | ||
| + | dot("$D$", D, NW); | ||
| + | dot("$E$", E, NW); | ||
| + | dot("$F$", F, SW); | ||
| + | dot("$G$", G, S); | ||
| + | dot("$H$", H, N); | ||
| + | dot("$I$", I, NE); | ||
| + | dot("$J$", J, SE); | ||
| + | </asy> | ||
Latest revision as of 09:59, 16 July 2024
Problem
Squares 
, 
, and 
are equal in area. Points 
and 
are the midpoints of sides 
and 
, respectively. What is the ratio of the area of the shaded pentagon 
to the sum of the areas of the three squares?
![[asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.5,2); B = (1.5,2); C = (1.5,1); D = (0.5,1); E = (0,1); F = (0,0); G = (1,0); H = (1,1); I = (2,1); J = (2,0); draw(A--B); draw(C--B); draw(D--A); draw(F--E); draw(I--J); draw(J--F); draw(G--H); draw(A--J); filldraw(A--B--C--I--J--cycle,grey); draw(E--I); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, NW); dot("$F$", F, SW); dot("$G$", G, S); dot("$H$", H, N); dot("$I$", I, NE); dot("$J$", J, SE); [/asy]](http://latex.artofproblemsolving.com/1/6/9/169139bda69e81f8342f246f95cc3f636c1f9abe.png)
