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 need to determine the slope of the line that passes through the points A (0, 12) and C (9, 0).

First, we calculate the slope \( m \) of the line using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, we have the coordinates of point A as \( (0, 12) \) and point C as \( (9, 0) \).

Substituting the coordinates into the formula, we have:

\[
m = \frac{0 - 12}{9 - 0} = \frac{-12}{9} = -\frac{4}{3}
\]

Now we have the slope of line AC, which is \( -\frac{4}{3} \).

Next, we can use the point-slope form of the equation of a line to find the equation of line AC. The equation in point-slope form is given by:

\[
y - y_1 = m(x - x_1)
\]

Using point A \( (0, 12) \):

\[
y - 12 = -\frac{4}{3}(x - 0)
\]

This simplifies to:

\[
y - 12 = -\frac{4}{3}x
\]

Now, we can add 12 to both sides to write it in slope-intercept form:

\[
y = -\frac{4}{3}x + 12
\]

Now, we need to find the x-coordinate of point D where \( y = 4 \).

Substituting \( y = 4 \) into the equation:

\[
4 = -\frac{4}{3}x + 12
\]

Next, we will isolate \( x \):

1. Subtract 12 from both sides:

\[
4 - 12 = -\frac{4}{3}x
\]
\[
-8 = -\frac{4}{3}x
\]

2. Multiply both sides by \( -\frac{3}{4} \):

\[
x = -8 \cdot -\frac{3}{4}
\]
\[
x = 6
\]

Therefore, the value of \( x \) for point D is:

\[
\boxed{6}
\]