Isosceles triangle $P Q R$ has $P Q=P R$ and $Q R=300$. Point $S$ is on $P Q$ and $T$ is on $P R$ so that $S T$ is perpendicular to $P R, S T=120, T R=271$, and $Q S=221$. The area of quadrilateral $S T R Q$ is
(A) 21275
(B) 40605
(C) 46860
(D) 54000
(E) 54603

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-064.jpg?height=439&width=379&top_left_y=1466&top_left_x=1293)

## Part C: Each correct answer is worth 8.
We calculate the area of quadrilateral $S T R Q$ by subtracting the area of $\triangle P T S$ from the area of $\triangle P Q R$.

Let $P T=x$.

Then $P R=P T+T R=x+271$.

Since $P Q=P R=x+271$ and $S Q=221$, then $P S=P Q-S Q=(x+271)-221=x+50$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-126.jpg?height=433&width=379&top_left_y=1369&top_left_x=927)

By the Pythagorean Theorem in $\triangle P T S$, we have

$$
\begin{aligned}
P T^{2}+T S^{2} & =P S^{2} \\
x^{2}+120^{2} & =(x+50)^{2} \\
x^{2}+14400 & =x^{2}+100 x+2500 \\
11900 & =100 x \\
x & =119
\end{aligned}
$$

Therefore, $\triangle P T S$ has $P T=x=119, T S=120$, and $P S=x+50=169$.

Since $\triangle P T S$ is right-angled at $T$, then its area is $\frac{1}{2}(P T)(T S)=\frac{1}{2}(119)(120)=7140$.

Furthermore, in $\triangle P Q R$, we have $P R=P Q=x+271=390$.

Now, $\triangle P Q R$ is isosceles, so when we draw a median $P X$ from $P$ to the midpoint $X$ of $Q R$, it is perpendicular to $Q R$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-127.jpg?height=442&width=396&top_left_y=153&top_left_x=927)

Since $X$ is the midpoint of $Q R$ and $Q R=300$, then $Q X=\frac{1}{2} Q R=150$.

We can use the Pythagorean Theorem in $\triangle P X Q$ to conclude that

$$
P X=\sqrt{P Q^{2}-Q X^{2}}=\sqrt{390^{2}-150^{2}}=\sqrt{21600}=360
$$

since $P X>0$.

Since $P X$ is a height in $\triangle P Q R$, then the area of $\triangle P Q R$ is $\frac{1}{2}(Q R)(P X)=\frac{1}{2}(300)(360)=54000$.

Finally, the area of $S T R Q$ is the difference in the areas of these two triangles, or $54000-7140$, which equals 46860 .

ANSWER: (C)