To solve this problem, we need to find the point on the y-axis where a line, perpendicular to the given line, passes through the blue point on the graph.

Step 1: Find the Slope of the Given Line

First, determine the slope of the given orange line by observing two points it passes through. From the graph:

• The line passes through $(-10, -6)$ and $(0, 2)$.

Using the slope formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$ $\text{slope} = \frac{2 - (-6)}{0 - (-10)} = \frac{8}{10} = \frac{4}{5}$

Step 2: Find the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal. Therefore, if the slope of the given line is $\frac{4}{5}$, the slope of the perpendicular line is: $\text{slope of perpendicular line} = -\frac{5}{4}$

Step 3: Write the Equation of the Perpendicular Line

The equation of a line in point-slope form is: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is the point the line passes through.

From the graph, the coordinates of the blue point are $(2, -2)$. Using this point and the slope $-\frac{5}{4}$: $y - (-2) = -\frac{5}{4}(x - 2)$ $y + 2 = -\frac{5}{4}x + \frac{5}{2}$ $y = -\frac{5}{4}x + \frac{5}{2} - 2$ $y = -\frac{5}{4}x + \frac{1}{2}$

Step 4: Find the y-Intercept

To find the point where this line crosses the y-axis, set $x = 0$: $y = -\frac{5}{4}(0) + \frac{1}{2}$ $y = \frac{1}{2}$

The point on the y-axis is $(0, \frac{1}{2})$. However, this point does not match any of the given answer choices. It appears that the coordinates might need to be checked or simplified further, based on the available answer choices.

Upon review, the closest point based on the problem and the graph is not correctly listed among the multiple-choice options.

Would you like me to recheck any steps or provide further details?

Follow-up Questions

1. How do we determine the equation of a line using a slope and a point?
2. Why is the slope of a perpendicular line the negative reciprocal of the original slope?
3. What happens if two lines have the same slope?
4. How can you verify if two lines are truly perpendicular on a graph?
5. What is the significance of the y-intercept in the equation of a line?

Tip: Always double-check the coordinates of key points on the graph when determining slopes and intercepts to avoid calculation errors.