Question

which points could be on the line that is parallel to \\(\overleftrightarrow{gh}\\) and passes through point j? check all that apply. \\(\square\\) (-3, 5) \\(\square\\) (1, 5) \\(\square\\) (3, -2) \\(\square\\) (3, 2) \\(\square\\) (5, 1)

Explanation:

Step1: Find slope of $\overleftrightarrow{GH}$

First, identify coordinates: $G(-4,1)$, $H(2,-2)$.
Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m=\frac{-2-1}{2-(-4)}=\frac{-3}{6}=-\frac{1}{2}$

Step2: Write line equation for point J

Point $J(1,3)$, parallel lines have equal slope $m=-\frac{1}{2}$.
Use point-slope form: $y-y_1=m(x-x_1)$
$y-3=-\frac{1}{2}(x-1)$
Simplify to slope-intercept: $y=-\frac{1}{2}x+\frac{1}{2}+3=-\frac{1}{2}x+\frac{7}{2}$

Step3: Test each candidate point

1. For $(-3,5)$: $y=-\frac{1}{2}(-3)+\frac{7}{2}=\frac{3}{2}+\frac{7}{2}=5$ ✔️
2. For $(1,5)$: $y=-\frac{1}{2}(1)+\frac{7}{2}=3

eq5$ ❌

1. For $(3,-2)$: $y=-\frac{1}{2}(3)+\frac{7}{2}=2

eq-2$ ❌

1. For $(3,2)$: $y=-\frac{1}{2}(3)+\frac{7}{2}=2$ ✔️
2. For $(5,1)$: $y=-\frac{1}{2}(5)+\frac{7}{2}=1$ ✔️

Answer:

(-3, 5), (3, 2), (5, 1)