Question

Point E is drawn on the graph so that line EF is parallel to line CD.

On a coordinate plane, line C D goes through (negative 10, negative 2) and (10, 6). Point F is at (6, negative 4).

If the coordinates of point E are (, y), what is the value of y?

−8
−4
0
8

Answer 1

To find the value of \( y \) for point \( E \) such that line \( EF \) is parallel to line \( CD \), we first need to determine the slope of line \( CD \).

The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula:

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

Using the points given for line \( CD \): \( (−10, −2) \) and \( (10, 6) \):

1. Assign the points:
- \( (x_1, y_1) = (-10, -2) \)
- \( (x_2, y_2) = (10, 6) \)

2. Calculate the slope:
\[
m_{CD} = \frac{6 - (-2)}{10 - (-10)} = \frac{6 + 2}{10 + 10} = \frac{8}{20} = \frac{2}{5}
\]

Since line \( EF \) is parallel to line \( CD \), the slope of line \( EF \) will also be \( \frac{2}{5} \).

Now we find the slope of line \( EF \) using point \( F \) at \( (6, -4) \) and point \( E \) at \( (x, y) \):

\[
m_{EF} = \frac{y - (-4)}{x - 6} = \frac{y + 4}{x - 6}
\]

Setting the slopes equal since the lines are parallel:

\[
\frac{y + 4}{x - 6} = \frac{2}{5}
\]

Cross-multiplying gives:

\[
5(y + 4) = 2(x - 6)
\]

Expanding both sides:

\[
5y + 20 = 2x - 12
\]

Rearranging the equation gives:

\[
5y = 2x - 12 - 20
\]
\[
5y = 2x - 32
\]

Now solving for \( y \):

\[
y = \frac{2}{5}x - \frac{32}{5}
\]

At this point, we need a specific value for \( x \) to calculate \( y \). However, since we don't have \( x \), we can use point \( E \) to find a value of \( y \) given possible coordinates that would yield \( y \).

To check the provided options for \( y \):
- If \( y = -8 \):
\[
-8 = \frac{2}{5}x - \frac{32}{5} \quad \Rightarrow \quad -8 + \frac{32}{5} = \frac{2}{5}x
\]
\[
-\frac{40}{5} + \frac{32}{5} = \frac{2}{5}x \quad \Rightarrow \quad -\frac{8}{5} = \frac{2}{5}x \quad \Rightarrow \quad x = -4
\]

- If \( y = -4 \):
\[
-4 = \frac{2}{5}x - \frac{32}{5} \quad \Rightarrow \quad -4 + \frac{32}{5} = \frac{2}{5}x
\]
\[
-\frac{20}{5} + \frac{32}{5} = \frac{2}{5}x \quad \Rightarrow \quad \frac{12}{5} = \frac{2}{5}x \quad \Rightarrow \quad x = 6
\] (which is point \( F \))

- If \( y = 0 \):
\[
0 = \frac{2}{5}x - \frac{32}{5} \quad \Rightarrow \quad \frac{32}{5} = \frac{2}{5}x \quad \Rightarrow \quad x = 16
\]

- If \( y = 8 \):
\[
8 = \frac{2}{5}x - \frac{32}{5} \quad \Rightarrow \quad 8 + \frac{32}{5} = \frac{2}{5}x
\]
\[
\frac{40}{5} + \frac{32}{5} = \frac{2}{5}x \quad \Rightarrow \quad \frac{72}{5} = \frac{2}{5}x \quad \Rightarrow \quad x = 36
\]

Thus, in terms of the given options, the point \( E \) would yield the value of \( y \) as:

\[
\boxed{8}
\]