Question

What is the value of x for point D shown on the coordinate plane?

The graph shows a line on the coordinate plane that passes through point A at (0, 12), point D at (x, 4), and point C at (9, 0). A triangle is shown connecting another point, B, at (0, 4) with point A and point D.

A.
4

B.
5

C.
6

D.
8

Answer 1

To find the value of \( x \) for point D, we first need to determine the equation of the line that passes through points A (0, 12) and C (9, 0).

1. **Find the slope (m)** of the line between points A and C:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 12}{9 - 0} = \frac{-12}{9} = -\frac{4}{3}
\]

2. **Use the point-slope form of a line** to write the equation using point A (0, 12):
\[
y - 12 = -\frac{4}{3}(x - 0)
\]
Simplifying this gives:
\[
y - 12 = -\frac{4}{3}x \quad \Rightarrow \quad y = -\frac{4}{3}x + 12
\]

3. **Substituting y = 4** (the y-coordinate of point D) into the line equation to find \( x \):
\[
4 = -\frac{4}{3}x + 12
\]
Rearranging to solve for \( x \):
\[
-\frac{4}{3}x = 4 - 12
\]
\[
-\frac{4}{3}x = -8
\]
Multiplying both sides by \(-\frac{3}{4}\):
\[
x = -8 \cdot -\frac{3}{4} = 6
\]

Thus, the value of \( x \) for point D is \( 6 \).

The answer is **C. 6**.