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!