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Using Midpoint and Distance Formulas

Using Midpoint and Distance Formulas 1.15 - Solution

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

Let's start with the side QR,\overline{QR}, which we will treat as the width w.w.
QR=(x2x1)2+(y2y1)2QR = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }
QR=(-11)2+(33)2QR = \sqrt{\left({\color{#0000FF}{\text{-} 1}}-{\color{#009600}{1}}\right)^2 + \left({\color{#0000FF}{3}}-{\color{#009600}{3}}\right)^2}
Simplify
QR=(-2)2+02QR=\sqrt{(\text{-} 2)^2+0^2}
QR=4QR=\sqrt{4}
QR=2QR=2
Next up is calculating the width of the rectangle. Let's use QP\overline{QP} for the width l.l. Again, we calculate using the Distance Formula.
QP=(x2x1)2+(y2y1)2QP = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }
QP=(-1(-1))2+(3(-1))2QP = \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=(-1+1)2+(3+1)2QP=\sqrt{(\text{-}1+1)^2+(3+1)^2}
\(QP=\sqrt0^2+4^2}\)
QP=16QP=\sqrt{16}
QP=4QP=4
We can now calculate the rectangle's area using the formula A=lw.A=lw.
A=lwA=lw
A=24A={\color{#0000FF}{2}}\cdot {\color{#009600}{4}}
A=8A=8
The area of rectangle PQRSPQRS is 8 square units.8\text{ square units}.