Hello, Veronica1999!

1. The diagonal of a rectangle is 15 cm, and the perimeter is 38 cm.

What is the area?

Is it possible to find the area without finding the dimensions of the rectangle?

. . Yes! $\displaystyle \text{Let }L\text{ = length, }\,W\text{ = width.}$

Code:

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| 15 * |
W | * |
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L

$\displaystyle \text{The diagonal is 15: }\:L^2 + W^2 \:=\:15^2 \quad\Rightarrow\quad L^2 + W^2 \:=\:225 \;\;[1]$

$\displaystyle \text{The perimeter is 38: }\:2L + 2W \:=\:38 \quad\Rightarrow\quad L + W \:=\:19\;\;[2]$

$\displaystyle \text{Square [2]: }\:(L+W)^2 \:=\:19^2 \quad\Rightarrow\quad \underbrace{L^2 + W^2}_{\text{This is 225}} + 2LW \:=\:361$

. . $\displaystyle 225 + 2LW \:=\:361 \quad\Rightarrow\quad 2LW \:=\:136 \quad\Rightarrow\quad \boxed{LW \:=\:68}$