Question

Line p goes through the points (–5,9) and (1,3). Line q goes through the point (4,7) and is perpendicular to line p.
Graph line q.

Answer 1

To graph line \( q \), which is perpendicular to line \( p \), we first need to find the slope of line \( p \) using the two points it's given: \((-5, 9)\) and \((1, 3)\).

### Step 1: Find the slope of line \( p \).

The formula for finding the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Substituting in our points:

- \((x_1, y_1) = (-5, 9)\)
- \((x_2, y_2) = (1, 3)\)

\[
m = \frac{3 - 9}{1 - (-5)} = \frac{-6}{1 + 5} = \frac{-6}{6} = -1
\]

So, the slope of line \( p \) is \( -1 \).

### Step 2: Find the slope of line \( q \).

Since line \( q \) is perpendicular to line \( p \), we take the negative reciprocal of the slope of line \( p \):

\[
m_q = -\frac{1}{-1} = 1
\]

### Step 3: Write the equation of line \( q \).

Line \( q \) goes through the point \((4, 7)\) and has a slope of \( 1 \). We can use the point-slope form of the equation of a line:

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

Substituting in our point and slope:

\[
y - 7 = 1(x - 4)
\]

Simplifying this gives:

\[
y - 7 = x - 4 \implies y = x + 3
\]

### Step 4: Graph line \( q \).

To graph the line \( y = x + 3 \):

1. Start by plotting the y-intercept, which is \( (0, 3) \).
2. Use the slope, which is \( 1 \), to find another point. From \( (0, 3) \), move up 1 unit and right 1 unit to find the point \( (1, 4) \).
3. You can also use the point \( (4, 7) \) given in the problem, which is another point on line \( q \).

### Summary of Points
- Points on line \( q \): \( (0, 3) \), \( (1, 4) \), and \( (4, 7) \).

You can plot these points on a coordinate system and draw a straight line through them to represent line \( q \).