A line segment joins the points $P(-4,1)$ and $Q(1,-11)$. What is the length of $P Q$ ?
(A) 13
(B) 12
(C) 12.5
(D) 13.6
(E) 12.6
The horizontal line through $P$ intersects the vertical line through $Q$ at $R(1,1)$, as shown.

Joining $P, Q, R$ creates right-angled $\triangle P Q R$, with hypotenuse $P Q$. The $x$-coordinate of $P$ is -4 and the $x$-coordinate of $R$ is 1 , and so $P R$ has length $1-(-4)=5$ (since $P$ and $R$ have equal $y$ coordinates).

The $y$-coordinate of $Q$ is -11 and the $y$-coordinate of $R$ is 1 , and so $Q R$ has length $1-(-11)=12$ (since $Q$ and $R$ have equal $x$ coordinates).

Using the Pythagorean Theorem, $P Q^{2}=P R^{2}+Q R^{2}$ or $P Q^{2}=5^{2}+12^{2}=25+144=169$, and so $P Q=\sqrt{169}=13$

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-098.jpg?height=520&width=477&top_left_y=995&top_left_x=1561)
(since $P Q>0$ ).

(Alternatively, we could have drawn the vertical line through $P$ and the horizontal line through $Q$ which meet at $(-4,-11)$.)

ANSWER: (A)

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