Question

which point on the y - axis lies on the line that passes through point c and is perpendicular to line ab? (-6,0) (0,-6) (0,2) (2,0)

Explanation:

Step1: Find slope of line AB

Let's assume two - point form. If we have two points on line AB, say \(A(x_1,y_1)\) and \(B(x_2,y_2)\). From the graph, assume \(A(- 4,4)\) and \(B(2,-8)\). The slope \(m_{AB}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-8 - 4}{2+4}=\frac{-12}{6}=-2\).

Step2: Find slope of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is \(m'=-\frac{1}{m}\). So, if \(m_{AB}=-2\), the slope of the line perpendicular to AB, \(m = \frac{1}{2}\).

Step3: Find the equation of the line passing through point C

Assume point \(C(4,4)\). Using the point - slope form \(y - y_1=m(x - x_1)\), where \(x_1 = 4,y_1 = 4\) and \(m=\frac{1}{2}\). We get \(y - 4=\frac{1}{2}(x - 4)\), which simplifies to \(y-4=\frac{1}{2}x - 2\), or \(y=\frac{1}{2}x+2\).

Step4: Find the y - intercept

The line intersects the y - axis when \(x = 0\). Substitute \(x = 0\) into \(y=\frac{1}{2}x + 2\), we get \(y=2\). So the point on the y - axis is \((0,2)\).

Answer:

C. \((0,2)\)