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 \(B(0, - 4)\). Using slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we have \(m_{AB}=\frac{-4 - 4}{0+2}=\frac{-8}{2}=-4\).

Step2: Find slope of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is \(m'=-\frac{1}{m}\). So the slope of the 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 \(y - y_1=m'(x - x_1)\). Substituting \(m'=\frac{1}{4}\), \(x_1 = 0\) and \(y_1=2\), we get \(y - 2=\frac{1}{4}(x - 0)\), which simplifies to \(y=\frac{1}{4}x+2\).

Step4: Test each point

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

Answer:

\((-4,1)\)