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# Using Midpoint and Distance Formulas

## Using Midpoint and Distance Formulas 1.15 - Solution

The area of a rectangle is calculated using the formula \begin{aligned} A=\ell w \end{aligned} To calculate the area of the rectangle $PQRS,$ we will first need to determine its side lengths using the given points and the Distance Formula.

Let's start with the side $\overline{QR},$ which we will treat as the width $w.$
$QR = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }$
$QR = \sqrt{\left({\color{#0000FF}{\text{-} 1}}-{\color{#009600}{1}}\right)^2 + \left({\color{#0000FF}{3}}-{\color{#009600}{3}}\right)^2}$
Simplify
$QR=\sqrt{(\text{-} 2)^2+0^2}$
$QR=\sqrt{4}$
$QR=2$
Next up is calculating the width of the rectangle. Let's use $\overline{QP}$ for the width $l.$ Again, we calculate using the Distance Formula.
$QP = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }$
$QP = \sqrt{\left({\color{#0000FF}{\text{-}1}}-\left({\color{#009600}{\text{-} 1}}\right)\right)^2 + \left({\color{#0000FF}{3}}-\left({\color{#009600}{\text{-} 1}}\right)\right)^2}$
Simplify
$QP=\sqrt{(\text{-}1+1)^2+(3+1)^2}$
$$QP=\sqrt0^2+4^2}$$
$QP=\sqrt{16}$
$QP=4$
We can now calculate the rectangle's area using the formula $A=lw.$
$A=lw$
$A={\color{#0000FF}{2}}\cdot {\color{#009600}{4}}$
$A=8$
The area of rectangle $PQRS$ is $8\text{ square units}.$