In the diagram, what is the perimeter of polygon PQRST?
(A) 24
(B) 23
(C) 25
(D) 26
(E) 27

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-092.jpg?height=352&width=398&top_left_y=1540&top_left_x=1297)
We extend $Q R$ to meet $T S$ at $X$.

Since $P Q=Q R$, then $Q R=3$.

Since $P Q X T$ has three right angles, it must be a rectangle, so $T X=P Q=3$.

Also, $Q X=P T=6$.

Since $T S=7$ and $T X=3$, then $X S=T S-T X=7-3=4$.

Since $Q X=6$ and $Q R=3$, then $R X=Q X-Q R=6-3=3$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-176.jpg?height=347&width=393&top_left_y=2255&top_left_x=926)

Since $P Q X T$ is a rectangle, then $\angle R X S=90^{\circ}$.

By the Pythagorean Theorem in $\triangle R X S$,

$$
R S^{2}=R X^{2}+X S^{2}=3^{2}+4^{2}=9+16=25
$$

so $R S=5$, since $R S>0$.

Therefore, the perimeter is $P Q+Q R+R S+S T+T P=3+3+5+7+6=24$.

ANSWER: (A)