2022 AMC 10B Problem 2

The area of a rhombus (like any parallelogram) is base times height. The segment $BP$ is perpendicular to $AD$, so it is the height when $AD$ is the base. We need the lengths $AD$ and $BP$.

1. The base. $P$ lies on $AD$, so $AD = AP + PD = 3 + 2 = 5$.
2. Another side. All four sides of a rhombus are equal, so $AB = 5$ too.
3. The height. Triangle $ABP$ has a right angle at $P$, a long side $AB = 5$ and a leg $AP = 3$. By the Pythagorean theorem,
$$BP = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4.$$
(This is the familiar $3$-$4$-$5$ right triangle.)
4. The area. Base times height is $5 \cdot 4 = 20$.

The answer is $\boxed{\textbf{(D)}\ 20}$.

A line dropped from a corner straight down to the opposite side is exactly the height of a parallelogram. The fact that every side of a rhombus has the same length turns the pieces $3$ and $2$ into a right triangle whose long side you already know. When a right angle appears inside a rhombus, look for a right triangle that uses one full side of the rhombus as its longest side.

Since $P$ lies on $\overline{AD}$, the side $AD = AP + PD = 3 + 2 = 5$. A rhombus has four equal sides, so $AB = 5$ as well.

Triangle $ABP$ is right-angled at $P$ with hypotenuse $AB = 5$ and leg $AP = 3$, so
$$BP = \sqrt{5^2 - 3^2} = 4.$$

A rhombus is a parallelogram, and $BP$ is the height onto base $AD$. Therefore the area is $AD \cdot BP = 5 \cdot 4 = 20$.

The answer is $\boxed{\textbf{(D)}\ 20}$.