In the diagram, the lengths of four of the sides of the figure are shown in terms of $x$. Assuming that $x \neq 0$, the perimeter of the figure is
(A) $8 x$
(B) $9 x$
(C) $12 x$
(D) $11 x$
(E) $10 x$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-008.jpg?height=333&width=312&top_left_y=1980&top_left_x=1313)
Because all of the angles in the figure are right angles, each line segment is either horizontal or vertical.

The height of the figure is $3 x$ and the width of the figure is $2 x$.

This means that the length of the unmarked vertical segment must equal $3 x-x=2 x$.

Also, the length of the unmarked horizontal segment must equal $2 x-x=x$.

Starting in the top left corner and adding lengths in a clockwise direction, the perimeter is $x+2 x+x+x+2 x+3 x=10 x$.

Alternatively, we can "complete the rectangle" by sliding the shortest horizontal side and the shortest vertical side as shown to form a rectangle with height $3 x$ and width $2 x$ :

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-012.jpg?height=325&width=306&top_left_y=328&top_left_x=969)

The perimeter of this rectangle is $2 \times 2 x+2 \times 3 x=10 x$.

ANSWER: (E)