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Let's solve the problem step by step based on the image:

### Given:
1. The equation of the line is \( y = -4x + 32 \), which crosses the \( y \)-axis at point \( A \) and the \( x \)-axis at point \( B \).
2. A perpendicular line to this passes through point \( B \) and crosses the \( y \)-axis at point \( C \).
3. We are asked to find the area of triangle \( ABC \).

### Step 1: Find coordinates of point \( A \) (where the line crosses the \( y \)-axis)
When the line crosses the \( y \)-axis, \( x = 0 \).

Substitute \( x = 0 \) into the line equation:
\[
y = -4(0) + 32 = 32
\]
Thus, point \( A \) is \( (0, 32) \).

### Step 2: Find coordinates of point \( B \) (where the line crosses the \( x \)-axis)
When the line crosses the \( x \)-axis, \( y = 0 \).

Set \( y = 0 \) in the equation \( y = -4x + 32 \) and solve for \( x \):
\[
0 = -4x + 32
\]
\[
4x = 32 \implies x = 8
\]
Thus, point \( B \) is \( (8, 0) \).

### Step 3: Find the slope of the perpendicular line and coordinates of point \( C \)
The slope of the given line \( y = -4x + 32 \) is \( -4 \). The slope of the line perpendicular to this is the negative reciprocal of \( -4 \), which is \( \frac{1}{4} \).

This perpendicular line passes through point \( B(8, 0) \), so its equation is of the form:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = \frac{1}{4} \), \( (x_1, y_1) = (8, 0) \):
\[
y - 0 = \frac{1}{4}(x - 8)
\]
\[
y = \frac{1}{4}x - 2
\]
To find point \( C \), set \( x = 0 \) (since point \( C \) is on the \( y \)-axis):
\[
y = \frac{1}{4}(0) - 2 = -2
\]
Thus, point \( C \) is \( (0, -2) \).

### Step 4: Area of triangle \( ABC \)
The formula for the area of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substitute the coordinates \( A(0, 32) \), \( B(8, 0) \), and \( C(0, -2) \):
\[
\text{Area} = \frac{1}{2} \left| 0(0 - (-2)) + 8((-2) - 32) + 0(32 - 0) \right|
\]
\[
\text{Area} = \frac{1}{2} \left| 8(-34) \right| = \frac{1}{2} \times 272 = 136
\]

Thus, the area of triangle \( ABC \) is \( \boxed{136} \) square units.

Would you like further clarifications on any step?

### 5 Related Questions:
1. How do you find the area of a triangle using the determinant method?
2. What is the relationship between slopes of perpendicular lines?
3. How can we derive the equation of a line passing through two points?
4. How does changing the slope affect the position of the intercepts?
5. How do we calculate the distance between two points on a coordinate plane?

### Tip:
Always double-check the signs of coordinates when calculating areas to avoid errors with negative results!

The line y = -4x + 32 crosses the y and x axes at the points A and B. A line perpendicular to this, that passes through point B, crosses the y axis at point C. Determine the area of triangle ABC.

This problem involves finding the area of a triangle formed by the line y = -4x + 32 and a perpendicular line crossing the x and y axes. Learn how to use the equation of a line, slope relationships, and the triangle area formula. Suitable for high school students.

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