Question

which point is on the line that passes through point z and is perpendicular to line ab? (-4,1) (1,-2) (2,0) (4,4)

Explanation:

Step1: Find slope of line AB

Let \(A(-2,4)\) and \(B(0, - 4)\). Slope \(m_{AB}=\frac{y_B - y_A}{x_B - x_A}=\frac{-4 - 4}{0+2}=\frac{-8}{2}=-4\).

Step2: Find slope of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is \(m'=-\frac{1}{m}\). So slope of line perpendicular to \(AB\) is \(m'=\frac{1}{4}\). Point \(Z(0,2)\).

Step3: Use point - slope form

The point - slope form of a line is \(y - y_1=m'(x - x_1)\). Here \(x_1 = 0,y_1 = 2,m'=\frac{1}{4}\), so \(y-2=\frac{1}{4}(x - 0)\) or \(y=\frac{1}{4}x+2\).

Step4: Check each point

For point \((-4,1)\): \(y=\frac{1}{4}\times(-4)+2=-1 + 2=1\), satisfies the equation.
For point \((1,-2)\): \(y=\frac{1}{4}\times1+2=\frac{1 + 8}{4}=\frac{9}{4}
eq-2\).
For point \((2,0)\): \(y=\frac{1}{4}\times2+2=\frac{1}{2}+2=\frac{5}{2}
eq0\).
For point \((4,4)\): \(y=\frac{1}{4}\times4+2=1 + 2=3
eq4\).

Answer:

\((-4,1)\)