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? \\( \circ\\ (-3.6, 0) \\) \\( \circ\\ (-2, 0) \\) \\( \circ\\ (0, -3.6) \\) \\( \circ\\ (0, -2) \\)

Explanation:

Step1: Find slope of given line

The given line passes through \((0, 6)\) and \((-6, -4)\) (wait, no, looking at the graph, the orange points: one is \((0,6)\) and another is \((-6, -4)\)? Wait, no, let's check coordinates. Wait, the blue point is at \((4, -6)\)? Wait, no, the grid: each square is 1 unit. Let's take two points on the orange line: \((0, 6)\) and \((-4, -4)\)? Wait, no, the left orange point: x=-6, y=-4? Wait, no, let's calculate slope correctly. Let's take \((0, 6)\) and \((-6, -4)\)? Wait, no, from (0,6) to (-6, -4): change in y is -4 -6 = -10, change in x is -6 -0 = -6, so slope is \(\frac{-10}{-6}=\frac{5}{3}\)? Wait, no, maybe I misread. Wait, the line goes through (0,6) and (-4, -4)? No, let's look again. Wait, the orange line: when x=0, y=6; when x=-6, y=-4? Wait, no, the left orange dot: x=-6, y=-4? Wait, no, the grid: x from -10 to 10, y from -10 to 10. The orange line passes through (0,6) and (-4, -4)? Wait, no, the distance between (0,6) and (-4, -4): no, maybe (0,6) and (-6, -4) is wrong. Wait, let's take two points: (0,6) and (-6, -4) – no, the slope would be (6 - (-4))/(0 - (-6)) = 10/6 = 5/3. Wait, but the blue point is at (4, -6). Now, the perpendicular line has slope negative reciprocal of 5/3, so -3/5.

Step2: Equation of perpendicular line

The perpendicular line passes through (4, -6) (blue point) and has slope -3/5. So equation is \(y - (-6) = -\frac{3}{5}(x - 4)\), so \(y + 6 = -\frac{3}{5}x + \frac{12}{5}\). Then \(y = -\frac{3}{5}x + \frac{12}{5} - 6 = -\frac{3}{5}x + \frac{12}{5} - \frac{30}{5} = -\frac{3}{5}x - \frac{18}{5}\).

Step3: Find intersection with y-axis (x=0)

Substitute x=0 into the perpendicular line equation: \(y = -\frac{3}{5}(0) - \frac{18}{5} = -\frac{18}{5} = -3.6\). So the point is (0, -3.6).

Answer:

(0, -3.6)