In the diagram, $Y Q Z C$ is a rectangle with $Y C=8$ and $C Z=15$. Equilateral triangles $A B C$ and $P Q R$, each with side length 9, are positioned as shown with $R$ and $B$ on sides $Y Q$ and $C Z$, respectively. The length of $A P$ is
(A) 10
(B) $\sqrt{117}$
(C) 9
(D) 8
(E) $\sqrt{72}$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-130.jpg?height=474&width=412&top_left_y=1297&top_left_x=1323)

## Part C: Each correct answer is worth 8.
Since $\triangle A B C$ and $\triangle P Q R$ are both equilateral triangles with side length 9 , they are congruent triangles.

So we can think of translating $\triangle A B C$ until it lies on top of $\triangle P Q R$ by shifting it up 8 units, until $C B$ lies on $Y Q(C$ will coincide with $Y$ ) and then shifting it to the right $15-9=6$ units until $B$ coincides with $Q$. (This last shift is 6 units to the right since $Y Q=15$ and $C B=9$, so $B$ will be a distance of 6 from $Q$ before this second shift.) So we move from $A$ to $P$ by moving 8 units up and 6 to the right, or a distance of $\sqrt{6^{2}+8^{2}}=10$ units.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-253.jpg?height=545&width=463&top_left_y=1728&top_left_x=1362)

ANSWER: (A)