Question

identifying a point on perpendicular lines
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 \( M(2, 3) \) and \( N(-3, 2) \). Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
\( m_{MN} = \frac{2 - 3}{-3 - 2} = \frac{-1}{-5} = \frac{1}{5} \).

Step2: Find slope of perpendicular line

Perpendicular slope is negative reciprocal: \( m_{\perp} = -5 \).

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

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

Step4: Test each option

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

eq -12 \).

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

eq 8 \).

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

eq 13 \).

Answer:

(2, 2)