In the drawing below, three buildings have been constructed on a plot of land divided into rectangular lots. The perimeters of buildings $A$ and $B$ are $400 \mathrm{~m}$ and $240 \mathrm{~m}$, respectively. What is the perimeter of building $C$?

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|  | $A$ |  | $B$ |  |  | $C$ |  |
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Solution

In a plot, we have three important dimensions: the width $l$, the height $h$, and the diagonal $d$.

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-22.jpg?height=174&width=280&top_left_y=1758&top_left_x=794)

Let's call the perimeters of buildings $A, B$, and $C$ as $P_{A}, P_{B}$, and $P_{C}$, respectively. Their values are:

$$
\begin{aligned}
P_{A} & =4 d+2 l+4 h \\
& =400 \\
P_{B} & =2 d+2 l+2 h \\
& =240 \\
P_{C} & =3 d+l+3 h
\end{aligned}
$$

Dividing the first equation by 2, we get $2 d+l+2 h=200$. Subtracting the second value from this, we obtain

$$
\begin{aligned}
240-200 & =(2 d+2 l+2 h)-(2 d+l+2 h) \\
40 & =l
\end{aligned}
$$

Analyzing the perimeter $P_{B}$, we have $d+l+h=120$. Therefore, $d+h=120-l=80$ and finally

$$
\begin{aligned}
P_{C} & =3(d+h)+l \\
& =3 \cdot 80+40 \\
& =240
\end{aligned}
$$

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