In the figure shown, the outer square has an area of $9 \mathrm{~cm}^{2}$, the inner square has an area of $1 \mathrm{~cm}^{2}$, and the four rectangles are identical. What is the perimeter of one of the four identical rectangles?
(A) $6 \mathrm{~cm}$
(B) $8 \mathrm{~cm}$
(C) $10 \mathrm{~cm}$
(D) $9 \mathrm{~cm}$
(E) $7 \mathrm{~cm}$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-041.jpg?height=301&width=298&top_left_y=2229&top_left_x=1315)

Grade 7
Solution 1

The outer square has an area of $9 \mathrm{~cm}^{2}$, so the sides of this outer square have length $3 \mathrm{~cm}$ (since $3 \times 3=9$ ), and thus $P N=3 \mathrm{~cm}$. The inner square has an area of $1 \mathrm{~cm}^{2}$, so the sides of this inner square have length $1 \mathrm{~cm}$ (since $1 \times 1=1$ ), and thus $M R=1 \mathrm{~cm}$. Since $P N=3 \mathrm{~cm}$, then $P S+S N=3 \mathrm{~cm}$ and so $Q R+S N=3 \mathrm{~cm}$ (since $Q R=P S$ ).

But $Q R=Q M+M R$, so then $Q M+M R+S N=3 \mathrm{~cm}$ or $Q M+1+S N=3 \mathrm{~cm}$ (since $M R=1 \mathrm{~cm})$.

From this last equation we get $Q M+S N=2 \mathrm{~cm}$.

Since each of $Q M$ and $S N$ is the width of an identical rectangle, then $Q M=S N=1 \mathrm{~cm}$.

Using $P S+S N=3 \mathrm{~cm}$, we get $P S+1=3 \mathrm{~cm}$ and so $P S=2 \mathrm{~cm}$.

Since the rectangles are identical, then $S N=P Q=1 \mathrm{~cm}$.

The perimeter of rectangle $P Q R S$ is $2 \times(P S+P Q)=2 \times(2+1)=2 \times 3=6 \mathrm{~cm}$.

## Solution 2

The outer square has an area of $9 \mathrm{~cm}^{2}$, so the sides of this outer square have length $3 \mathrm{~cm}$ (since $3 \times 3=9$ ), and thus $P N=3 \mathrm{~cm}$.

Since $P N=3 \mathrm{~cm}$, then $P S+S N=3 \mathrm{~cm}$.

Since each of $P Q$ and $S N$ is the width of an identical rectangle, then $P Q=S N$ and so $P S+S N=P S+P Q=3 \mathrm{~cm}$.

The perimeter of $P Q R S$ is $2 \times(P S+P Q)=2 \times 3=6 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-189.jpg?height=396&width=355&top_left_y=1561&top_left_x=1576)

ANsWER: (A)