Question

which point could be on the line that is perpendicular to \\(\overleftrightarrow{mn}\\) and passes through point k? \\(\bigcirc\\) (0, -12) \\(\bigcirc\\) (2, 2) \\(\bigcirc\\) (4, 8) \\(\bigcirc\\) (5, 13)

Explanation:

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

Points \( M(2, 3) \), \( N(-3, 2) \). Slope \( m_{MN}=\frac{3 - 2}{2 - (-3)}=\frac{1}{5} \).

Step2: Find slope of perpendicular line

Perpendicular slope \( m = -5 \) (negative reciprocal of \( \frac{1}{5} \)).

Step3: Find equation of line through \( K(3, -3) \)

Using point - slope form \( y - y_1 = m(x - x_1) \), \( y - (-3)=-5(x - 3) \), so \( y + 3=-5x + 15 \), \( y=-5x + 12 \).

Step4: Test each option

• For \( (0, - 12) \): \( y=-5(0)+12 = 12

eq - 12 \).

• For \( (2, 2) \): \( y=-5(2)+12 = 2 \). This works.
• For \( (4, 8) \): \( y=-5(4)+12=-8

eq8 \).

• For \( (5, 13) \): \( y=-5(5)+12=-13

eq13 \).

Answer:

B. (2, 2)