Question

which point is on the line that passes through point h and is perpendicular to line fg? (-6, 10) (-2, -12) (0, -2) (4, 2)

Explanation:

Step1: Find slope of line FG

First, identify coordinates: $F(-10,-8)$, $G(8,4)$.
Slope formula: $m_{FG}=\frac{y_2-y_1}{x_2-x_1}$
$\displaystyle m_{FG}=\frac{4-(-8)}{8-(-10)}=\frac{12}{18}=\frac{2}{3}$

Step2: Find perpendicular slope

Perpendicular slope is negative reciprocal:
$m_{\perp}=-\frac{3}{2}$

Step3: Find equation of target line

Identify $H(8,-6)$. Use point-slope form $y-y_1=m(x-x_1)$
$y-(-6)=-\frac{3}{2}(x-8)$
Simplify to slope-intercept:
$y+6=-\frac{3}{2}x+12$
$y=-\frac{3}{2}x+6$

Step4: Test each option

Test $(-6,10)$:
$y=-\frac{3}{2}(-6)+6=9+6=15
eq10$
Test $(-2,-12)$:
$y=-\frac{3}{2}(-2)+6=3+6=9
eq-12$
Test $(0,-2)$:
$y=-\frac{3}{2}(0)+6=6
eq-2$
Test $(4,2)$:
$y=-\frac{3}{2}(4)+6=-6+6=2$ (matches)

Answer:

(4, 2)