Question

which point is on the line that passes through point z and is perpendicular to line ab? \\( \bigcirc\\ (-4, 1) \\) \\( \bigcirc\\ (1, -2) \\) \\( \bigcirc\\ (2, 0) \\) \\( \bigcirc\\ (4, 4) \\) (accompanied by a coordinate grid image with points a, b, z)

Explanation:

Step1: Find slope of AB

Points A(-2,4) and B(0,-4). Slope \( m_{AB} = \frac{-4 - 4}{0 - (-2)} = \frac{-8}{2} = -4 \).

Step2: Find slope of perpendicular line

Perpendicular slope \( m = \frac{1}{4} \) (negative reciprocal of -4).

Step3: Equation of line through Z(0,2)

Using point - slope form \( y - y_1 = m(x - x_1) \), we get \( y - 2=\frac{1}{4}(x - 0) \), so \( y=\frac{1}{4}x + 2 \).

Step4: Check each option

• For (-4,1): \( y=\frac{1}{4}(-4)+2=-1 + 2 = 1 \). Matches.
• For (1,-2): \( y=\frac{1}{4}(1)+2=\frac{1}{4}+2=\frac{9}{4}

eq - 2 \).

• For (2,0): \( y=\frac{1}{4}(2)+2=\frac{1}{2}+2=\frac{5}{2}

eq0 \).

• For (4,4): \( y=\frac{1}{4}(4)+2=1 + 2 = 3

eq4 \).

Answer:

(-4, 1)