In the diagram, $\triangle P Q R$ is isosceles with $P Q=P R$ and $\triangle P R S$ is isosceles with $P S=S R=x$. Also, the perimeter of $\triangle P Q R$ is 22 , the perimeter of $\triangle P R S$ is 22 , and the perimeter of $P Q R S$ is 24 . What is the value of $x$ ?
(A) 7.5
(B) 6.5
(C) 7
(D) 6
(E) 8

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-051.jpg?height=334&width=330&top_left_y=2229&top_left_x=1342)
Since $P S=S R=x$ and the perimeter of $\triangle P R S$ is 22 , then $P R=22-P S-S R=22-2 x$.

Since $P Q=P R$ and $P R=22-2 x$, then $P Q=22-2 x$.

Since $\triangle P Q R$ has perimeter 22 , then $R Q=22-P R-P Q=22-(22-2 x)-(22-2 x)=4 x-22$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-102.jpg?height=353&width=415&top_left_y=1737&top_left_x=909)

Since the perimeter of $P Q R S$ is 24 , then

$$
\begin{aligned}
P Q+R Q+S R+P S & =24 \\
(22-2 x)+(4 x-22)+x+x & =24 \\
4 x & =24 \\
x & =6
\end{aligned}
$$

Therefore, $x=6$.

ANswer: (D)