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

Solution:

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

The four corner triangles have total area $2\cdot\frac12\cdot3\cdot4$ $+2\cdot\frac12\cdot6\cdot5$ $=12+30$ $=42$ . 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

Solution:

Problem 24 in Other Years