3. show that quadrilateral fghj is a parallelogram by showing that one pair of opposite sides are both parallel and congruent.
f(-4, -2), g(-2, 2), h(4, 3), & j(2, -1)

Question

3. show that quadrilateral fghj is a parallelogram by showing that one pair of opposite sides are both parallel and congruent.
    f(-4, -2), g(-2, 2), h(4, 3), & j(2, -1)

Accepted Answer

# Explanation:
## Step1: Calculate the length of $FG$ and $HJ$
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $F(-4,-2)$ and $G(-2,2)$:
$$
\begin{align*}
FG&=\sqrt{(-2+4)^2+(2 + 2)^2}\
&=\sqrt{(2)^2+(4)^2}\
&=\sqrt{4 + 16}\
&=\sqrt{20}=2\sqrt{5}
\end{align*}
$$
For $H(4,3)$ and $J(2,-1)$:
$$
\begin{align*}
HJ&=\sqrt{(2 - 4)^2+(-1 - 3)^2}\
&=\sqrt{(-2)^2+(-4)^2}\
&=\sqrt{4+16}\
&=\sqrt{20}=2\sqrt{5}
\end{align*}
$$
## Step2: Calculate the slope of $FG$ and $HJ$
Use the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For $F(-4,-2)$ and $G(-2,2)$:
$$m_{FG}=\frac{2+2}{-2 + 4}=\frac{4}{2}=2$$
For $H(4,3)$ and $J(2,-1)$:
$$m_{HJ}=\frac{-1 - 3}{2 - 4}=\frac{-4}{-2}=2$$

Since $FG = HJ=2\sqrt{5}$ and $m_{FG}=m_{HJ} = 2$, one pair of opposite sides ($FG$ and $HJ$) are both congruent and parallel.

# Answer:
Quadrilateral $FGHJ$ is a parallelogram because $FG\parallel HJ$ (slope of $FG$ = slope of $HJ = 2$) and $FG=HJ = 2\sqrt{5}$ (using distance formula).

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of \(FG\) and \(HJ\)

Step2: Calculate the slope of \(FG\) and \(HJ\)

Answer: