Rectangle $P Q R S$ is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. The length of $Q S$ is closest to
(A) 18
(B) 13
(C) 26
(D) 24
(E) 17

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-064.jpg?height=236&width=461&top_left_y=267&top_left_x=1298)
Let $s$ be the side length of each of the 60 identical squares.

Since the diagonal of each of the squares has length 2, then by the Pythagorean Theorem, $s^{2}+s^{2}=2^{2}$ or $2 s^{2}=4$, which gives $s^{2}=2$ or $s=\sqrt{2}$, since $s>0$.

Now $P Q=5 s$ and $P S=12 s$, so since $Q S>0$, then by the Pythagorean Theorem,

$$
Q S=\sqrt{P Q^{2}+P S^{2}}=\sqrt{(5 s)^{2}+(12 s)^{2}}=\sqrt{25 s^{2}+144 s^{2}}=\sqrt{169 s^{2}}=13 s
$$

Since $Q S=13 s$ and $s=\sqrt{2}$, then $Q S=13 \sqrt{2} \approx 18.38$.

Of the given choices, this is closest to 18 .

ANSWER: (A)

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