In the diagram, points $S$ and $T$ are on sides $Q R$ and $P Q$, respectively, of $\triangle P Q R$ so that $P S$ is perpendicular to $Q R$ and $R T$ is perpendicular to $P Q$. If $P T=1, T Q=4$, and $Q S=3$, what is the length of $S R$ ?
(A) 3
(B) $\frac{11}{3}$
(C) $\frac{15}{4}$
(D) $\frac{7}{2}$
(E) 4

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-022.jpg?height=341&width=455&top_left_y=2239&top_left_x=1255)
Since $P T=1$ and $T Q=4$, then $P Q=P T+T Q=1+4=5$.

$\triangle P S Q$ is right-angled at $S$ and has hypotenuse $P Q$.

We can thus apply the Pythagorean Theorem to obtain $P S^{2}=P Q^{2}-Q S^{2}=5^{2}-3^{2}=16$.

Since $P S>0$, then $P S=4$.

Consider $\triangle P S Q$ and $\triangle R T Q$.

Each is right-angled and they share a common angle at $Q$. Thus, these two triangles are similar.

This tells us that $\frac{P Q}{Q S}=\frac{Q R}{T Q}$.

Using the lengths that we know, $\frac{5}{3}=\frac{Q R}{4}$ and so $Q R=\frac{4 \cdot 5}{3}=\frac{20}{3}$.

Finally, $S R=Q R-Q S=\frac{20}{3}-3=\frac{11}{3}$.

ANSWER: (B)