# 1991 AHSME Problems/Problem 23

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## Problem

$[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle,dot); draw((2,2)--(0,0)--(0,1)--cycle,dot); draw((0,2)--(1,0),dot); MP("B",(0,0),SW);MP("A",(0,2),NW);MP("D",(2,2),NE);MP("C",(2,0),SE); MP("E",(0,1),W);MP("F",(1,0),S);MP("H",(2/3,2/3),E);MP("I",(2/5,6/5),N); dot((1,0));dot((0,1));dot((2/3,2/3));dot((2/5,6/5)); [/asy]$

If $ABCD$ is a $2\times2$ square, $E$ is the midpoint of $\overline{AB}$,$F$ is the midpoint of $\overline{BC}$,$\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is

$\text{(A) } \frac{1}{3}\quad \text{(B) } \frac{2}{5}\quad \text{(C) } \frac{7}{15}\quad \text{(D) } \frac{8}{15}\quad \text{(E) } \frac{3}{5}$

## Solution 1: Coordinate Geometry

Solution by e_power_pi_times_i

First, we find out the coordinates of the vertices of quadrilateral $BEIH$, then use the Shoelace Theorem to solve for the area. Denote $B$ as $(0,0)$. Then $E (0,1)$. Since I is the intersection between lines $DE$ and $AF$, and since the equations of those lines are $y = \dfrac{1}{2}x + 1$ and $y = -2x + 2$, $I (\dfrac{2}{5}, \dfrac{6}{5})$. Using the same method, the equation of line $BD$ is $y = x$, so $H (\dfrac{2}{3}, \dfrac{2}{3})$. Using the Shoelace Theorem, the area of $BEIH$ is $\dfrac{1}{2}\cdot\dfrac{14}{15} = \boxed{\textbf{(C) } \dfrac{7}{15}}$.

## See also

 1991 AHSME (Problems • Answer Key • Resources) Preceded byProblem 22 Followed byProblem 24 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 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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