Question

use the given line and the point not on the line to answer the question. what is the point on the line perpendicular to the given line, passing through the given point that is also on the y-axis?
(-3.6, 0)
(-2, 0)
(0, -3.6)
(0, -2)

Explanation:

Step1: Find slope of given line

Two points on given line: \((0, 6)\) and \((-6, -4)\). Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}=\frac{-4 - 6}{-6 - 0}=\frac{-10}{-6}=\frac{5}{3}\).

Step2: Slope of perpendicular line

Perpendicular slope \(m_{\perp}=-\frac{3}{5}\) (negative reciprocal).

Step3: Identify given point

Given point (not on line) is \((4, -6)\) (from graph: blue dot at \(x = 4\), \(y=-6\)).

Step4: Equation of perpendicular line

Using point - slope form \(y - y_1 = m_{\perp}(x - x_1)\): \(y+6 = -\frac{3}{5}(x - 4)\).

Step5: Find y - axis intersection (\(x = 0\))

Substitute \(x = 0\) into equation: \(y+6=-\frac{3}{5}(0 - 4)=-\frac{3}{5}(-4)=\frac{12}{5}=2.4\)? Wait, no, wait. Wait, maybe I misread the given point. Wait, looking at the graph, the blue dot is at \((4, -6)\)? Wait, no, maybe the given point is \((4, -6)\)? Wait, no, let's re - check. Wait, the two orange points: one at \((0,6)\) and one at \((-6, -4)\)? Wait, no, when \(x=-6\), \(y=-4\)? Wait, no, the lower orange point: let's count the grid. From \((0,6)\), moving left 6 units (x=-6) and down 10 units? No, wait, the line passes through \((0,6)\) and \((-4, -4)\)? Wait, no, the left orange dot: x=-6? Wait, no, the x - coordinate of the left orange dot: looking at the grid, from the origin (0,0), left 6 units is x=-6, and y=-4? Wait, no, the distance between (0,6) and (-6, -4): the change in y is - 10, change in x is - 6, so slope is \(\frac{-10}{-6}=\frac{5}{3}\), that's correct. Now, the blue dot: x = 4, y=-6 (since it's 4 units right of origin, 6 units down). Now, equation of perpendicular line: \(y - (-6)=-\frac{3}{5}(x - 4)\), so \(y + 6=-\frac{3}{5}x+\frac{12}{5}\). Now, find where x = 0 (y - axis). Substitute x = 0: \(y+6=\frac{12}{5}\), \(y=\frac{12}{5}-6=\frac{12 - 30}{5}=-\frac{18}{5}=-3.6\). So the point is \((0, - 3.6)\).

Answer:

\((0, - 3.6)\)