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To solve the problem, let's break it down step by step.

1. **Identify the slope of the given line:**
   - The given line passes through two points that can be identified by looking at the graph. It passes through points \((-8, 0)\) and \((0, 2)\).
   - The slope \(m\) of a line is calculated by the formula: 
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the coordinates of the points:
   \[
   m = \frac{2 - 0}{0 - (-8)} = \frac{2}{8} = \frac{1}{4}
   \]
   So, the slope of the given line is \( \frac{1}{4} \).

2. **Determine the slope of the perpendicular line:**
   - The slopes of two perpendicular lines are negative reciprocals of each other. So, if the slope of the given line is \( \frac{1}{4} \), the slope of the perpendicular line will be:
   \[
   m_{\text{perpendicular}} = -4
   \]

3. **Use the point-slope form of the equation of a 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 a point on the line.
   - We are given that the line passes through the point \((4, 5)\), and we know the slope of the perpendicular line is \(-4\).
   Substituting these values into the point-slope form:
   \[
   y - 5 = -4(x - 4)
   \]

4. **Convert to slope-intercept form:**
   - Expand and simplify the equation to get it into the slope-intercept form \(y = mx + b\):
   \[
   y - 5 = -4x + 16
   \]
   \[
   y = -4x + 16 + 5
   \]
   \[
   y = -4x + 21
   \]

Thus, the equation of the line perpendicular to the given line and passing through the point \((4, 5)\) is:
\[
y = -4x + 21
\]

Would you like more details on this or have any questions?

Here are five related questions for practice:
1. How do you find the equation of a line parallel to a given line?
2. What is the slope of a line perpendicular to one with slope \( \frac{-3}{5} \)?
3. How would the equation change if the perpendicular line passed through a different point?
4. What is the relationship between slopes of parallel lines?
5. How do you determine the y-intercept from an equation in slope-intercept form?

**Tip:** Always check for the negative reciprocal when dealing with perpendicular lines!

Write an equation of a line in slope-intercept form perpendicular to the line shown on the graph and that passes through the point (4, 5).

Learn how to find the equation of a line perpendicular to a given line on a graph and passing through the point (4, 5). This problem covers key algebraic concepts, including the slope formula and slope-intercept form, suitable for students in Grades 8-10.

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