Question

In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $\overline{HE}$ and $\overline{FG}?$

$6.8$

$7.1$

$7.6$

$7.8$

$8.1$

Answer: C

Concepts:parallelogram area decomposition Pythagorean Theorem

Difficulty rating: 1540

Small Hint:

Find the area of the rectangle and subtract the four corner triangles.

Big Hint:

Use the Pythagorean theorem to get the parallelogram base $HE.$

Solution:

The rectangle has side lengths $10$ and $8,$ so its area is $80.$

The four corner triangles have total area $\begin{aligned} &2\cdot\frac12\cdot3\cdot4 \\ &\quad{}+2\cdot\frac12\cdot6\cdot5 \\ &=12+30=42. \end{aligned}$ Thus parallelogram $EFGH$ has area $80-42=38.$

Segment $HE$ has length $5$ by the $3-4-5$ right triangle. Since the parallelogram area is $HE\cdot d,$ we have $5d=38,$ so $d=7.6.$

Thus, C is the correct answer.

2004 AMC 8 Problem 24

Answer: C

Difficulty rating: 1540

Small Hint:

Big Hint:

Solution:

Problem 24 in Other Years