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

Question

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.