Question

On a coordinate plane, line A B goes through (negative 2, 4) and (0, negative 4). Point Z is at (0, 2).
Which point is on the line that passes through point Z and is perpendicular to line AB?

(–4, 1)
(1, –2 )
(2, 0)
(4, 4)

Answer 1

To find the point on the line that passes through point Z (0, 2) and is perpendicular to line AB, we first need to determine the slope of line AB, and then find the slope of the perpendicular line.

1. **Calculate the slope of line AB**:
The formula for the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For points \( (-2, 4) \) and \( (0, -4) \):
\[
m_{AB} = \frac{-4 - 4}{0 - (-2)} = \frac{-8}{2} = -4
\]

2. **Determine the slope of the line that is perpendicular to AB**:
The slope of a line perpendicular to another line with slope \( m \) is given by:
\[
m_{perpendicular} = -\frac{1}{m}
\]
Thus, the slope of the line perpendicular to AB is:
\[
m_{perpendicular} = -\frac{1}{-4} = \frac{1}{4}
\]

3. **Write the equation of the line that passes through point Z (0, 2)**:
We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is point Z (0, 2) and \( m = \frac{1}{4} \):
\[
y - 2 = \frac{1}{4}(x - 0)
\]
Simplifying this, we get:
\[
y - 2 = \frac{1}{4}x
\]
\[
y = \frac{1}{4}x + 2
\]

4. **Check which of the given points lies on the line \( y = \frac{1}{4}x + 2 \)**:
We will substitute \( x \) values from the given points into the equation and see which one satisfies it.

- For \( (-4, 1) \):
\[
y = \frac{1}{4}(-4) + 2 = -1 + 2 = 1
\]
(This point satisfies the equation.)

- For \( (1, -2) \):
\[
y = \frac{1}{4}(1) + 2 = \frac{1}{4} + 2 = 2.25
\]
(This point does not satisfy the equation.)

- For \( (2, 0) \):
\[
y = \frac{1}{4}(2) + 2 = \frac{1}{2} + 2 = 2.5
\]
(This point does not satisfy the equation.)

- For \( (4, 4) \):
\[
y = \frac{1}{4}(4) + 2 = 1 + 2 = 3
\]
(This point does not satisfy the equation.)

Thus, the only point that lies on the line that is perpendicular to line AB and passes through point Z is:

\[
\boxed{(-4, 1)}
\]