Question

1. 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)

Explanation:

Step1: Calculate slope of FG

Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
$m_{FG} = \frac{2 - (-2)}{-2 - (-4)} = \frac{4}{2} = 2$

Step2: Calculate slope of HJ

$m_{HJ} = \frac{-1 - 3}{2 - 4} = \frac{-4}{-2} = 2$
(Slopes equal: FG || HJ)

Step3: Calculate length of FG

Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$FG = \sqrt{(-2 + 4)^2 + (2 + 2)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

Step4: Calculate length of HJ

$HJ = \sqrt{(2 - 4)^2 + (-1 - 3)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$
(Lengths equal: FG ≅ HJ)

Answer:

Since FG and HJ are both parallel and congruent, quadrilateral FGHJ is a parallelogram.