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.

Answer 1

To find the value of \( x \) for point D, we can first determine the equation of the line that passes through points A and C, and then we can use that equation to find the value of \( x \) when \( y = 4 \), since point D is at (x, 4).

1. **Identify Points A and C:**
- Point A is at \( (0, 12) \)
- Point C is at \( (9, 0) \)

2. **Calculate the slope (m) of line AC:**
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Applying this to points A and C:
\[
m = \frac{0 - 12}{9 - 0} = \frac{-12}{9} = -\frac{4}{3}
\]

3. **Use point-slope form to find the equation of the line:**
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Using point A (0, 12) and slope \( -\frac{4}{3} \):
\[
y - 12 = -\frac{4}{3}(x - 0)
\]
Simplifying this equation:
\[
y - 12 = -\frac{4}{3}x
\]
\[
y = -\frac{4}{3}x + 12
\]

4. **Find the value of \( x \) for point D where \( y = 4 \):**
We set \( y = 4 \) in the line equation:
\[
4 = -\frac{4}{3}x + 12
\]
Rearranging gives:
\[
-\frac{4}{3}x = 4 - 12
\]
\[
-\frac{4}{3}x = -8
\]
Multiplying both sides by \(-\frac{3}{4}\):
\[
x = -8 \cdot -\frac{3}{4} = 6
\]

5. **Conclusion:**
Thus, the value of \( x \) for point D is:
\[
\boxed{6}
\]