In the diagram, the horizontal distance between adjacent dots in the same row is 1 . Also, the vertical distance between adjacent dots in the same column is 1 . What is the perimeter of quadrilateral $P Q R S$ ?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-069.jpg?height=361&width=371&top_left_y=1950&top_left_x=1300)
The perimeter of quadrilateral $P Q R S$ equals $P Q+Q R+R S+S P$.

Since the dots are spaced 1 unit apart horizontally and vertically, then $P Q=4, Q R=4$, and $P S=1$.

Thus, the perimeter equals $4+4+R S+1$ which equals $R S+9$.

We need to determine the length of $R S$.

If we draw a horizontal line from $S$ to point $T$ on $Q R$, we create a right-angled triangle $S T R$ with $S T=4$ and $T R=3$.

By the Pythagorean Theorem, $R S^{2}=S T^{2}+T R^{2}=4^{2}+3^{2}=25$.

Since $R S>0$, then $R S=\sqrt{25}=5$.

Thus, the perimeter of quadrilateral $P Q R S$ is $5+9=14$.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-118.jpg?height=361&width=372&top_left_y=2042&top_left_x=1535)

ANSWER: (C)

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