1964 AHSME Problems/Problem 22

Problem

Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$ . Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$ . What is the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$ ?

$\textbf{(A)}\ 1:2 \qquad \textbf{(B)}\ 1:3 \qquad \textbf{(C)}\ 1:5 \qquad \textbf{(D)}\ 1:6 \qquad \textbf{(E)}\ 1:7$

Solution

If it works for a parallelogram $ABCD$ , it should also work for a unit square, with $A(0, 0), B(0, 1), C(1, 1), D(1, 0)$ . We are given that $E$ is the midpoint of $BD$ , so $E(0.5, 0.5)$ . If $F$ is on $DA$ , then $F(x, 0)$ . We note that $DF = 1-x$ and $DA = 1$ , so $DF = \frac{1}{3}DA$ means $1-x = \frac{1}{3}$ , or $x = \frac{2}{3}$ , and hence $F(\frac{2}{3}, 0)$ .

We note that $\triangle DFE$ has a base $DF$ that is $\frac{1}{3}$ and an altitude from $E$ to $DF$ that is $\frac{1}{2}$ . Therefore, $[DEF] = \frac{1}{2}bh = \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{12}$ .

Quadrilateral $ABEF$ can be split into $\triangle ABE$ and $\triangle AEF$ . The first triangle is $\frac{1}{4}$ of the unit square cut diagonally, so $[ABE] = \frac{1}{4}$ . The second triangle has base $AF$ that is $\frac{2}{3}$ and height $E$ to $AF$ that is $\frac{1}{2}$ . Therefore, $[AEF] = \frac{1}{2}bh = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{6}$ .

The entire quadrilateral $ABEF$ has area $\frac{1}{4} + \frac{1}{6} = \frac{5}{12}$ . This is $5$ times larger than the area of $\triangle DFE$ , so the ratio is $1:5$ , or $\boxed{\textbf{(C)}}$ .

Solution 2

$\begin{align*} \frac{A_{DFE}}{A_{ADE}} &= \frac{\frac{1}{3}DA}{DA} = \frac{1}{3} \\ A_{DFE} &= \frac{1}{3}A_{ADE} \\ A_{ADE} &= \frac{1}{2}A_{ADB} = \frac{1}{4}A_{ABCD} \\ \implies A_{DFE} &= \frac{1}{3} \cdot \frac{1}{4} A_{ABCD} = \frac{1}{12} A_{ABCD} \\ A_{ABEF} &= A_{ABD} - A_{DFE} \\ &= \frac{1}{2}A_{ABCD} - \frac{1}{12}A_{ABCD} \\ &= \frac{5}{12}A_{ABCD} \end{align*}$ Therefore, $\frac{A_{DFE}}{A_{ABEF}} = \frac{\frac{1}{12}}{\frac{5}{12}} = \frac{1}{5}$ , giving us the answer $\boxed{\textbf{(C)}}$ . -nullptr07

1964 AHSC (Problems • Answer Key • Resources)
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1964 AHSME Problems/Problem 22

Contents

Problem

Solution

Solution 2

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