Question

Let $ABCD$ be a rhombus with $AC = 16$ and $BD = 30.$ Let $N$ be a point on $\overline{AB},$ and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\overline{AC}$ and $\overline{BD},$ respectively. Which of the following is closest to the minimum possible value of $PQ?$

$6.5$

$6.75$

$7$

$7.25$

$7.5$

Answer: C

Concepts:rhombus altitude optimization

Difficulty rating: 2020

Solution:

Let $O$ be the intersection of the diagonals. Then $\triangle AOB$ is right-angled at $O$ with legs $OA = 8$ and $OB = 15.$ Quadrilateral $OPNQ$ has right angles at $O,$ $P,$ and $Q,$ so it is a rectangle and $PQ = ON.$

The minimum of $ON$ is the altitude from $O$ to $\overline{AB}$ in $\triangle AOB.$ Since $AB = \sqrt{8^2 + 15^2} = 17,$ equating the two area expressions gives $\begin{aligned} ON &= \frac{OA \cdot OB}{AB} \\ &= \frac{8 \cdot 15}{17} \\ &= \frac{120}{17} \approx 7.06. \end{aligned}$

This is closest to $7.$

Thus, the correct answer is C.

2003 AMC 12B Problem 22

Answer: C

Difficulty rating: 2020

Solution:

Problem 22 in Other Years