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# Parallel and Perpendicular Lines

## Key Concepts

• Write an equation of a line parallel to given line
• Understand the slope of perpendicular lines
• Write an equation of a line perpendicular to a given line

## Parallel and Perpendicular Lines

### Parallel Lines

Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect.

### Perpendicular Lines

Perpendicular lines are lines that intersect at a right (90 degrees) angle.

### Write an equation of a line Parallel to given line

Example 1:

What is the equation of the line in slope intercept form that passes through the point (-3, 5) and is parallel to the graph of y=4x -7?

Solution:

Step 1:

Identify the slope of the given line.

y = 4x – 7

The slope is 4. The slope of a parallel line will be the same.

Step 2:

Start with point form. Use the given point and the slope of the parallel line.

y- 5 = 4 (x+3)

y-5 = 4x +12

y= 4x +17

The equation of the line is y= 4x +17.

### Understand the slope of perpendicular lines

Why does it make sense that the slopes of perpendicular lines have opposite signs?

Solution:

Perpendicular lines are a bit more complicated.

If you visualize a line with a positive slope (so it’s an increasing line), then the perpendicular line must have a negative slope (because it will have to be a decreasing line).

So, perpendicular lines have slopes which have opposite signs.

Example:

Find the slope of a line perpendicular to the line y = −4x + 9.

They’ve given me the original line’s equation, and it’s in “y =” form, so it’s easy to find the slope.

I can just read the value of the equation: m = −4.

This slope can be turned into a fraction by putting it over 1, so this slope can be restated as:

To get the negative reciprocal, I need to flip this fraction and change the sign.

Then the slope of any line perpendicular to the given line is:

### Write an equation of a line perpendicular to a given line

Example: What is the equation to the line that passes through (6, -5) and is perpendicular to the graph of y=2x+3?

Solution:

Step 1:

Use the slope of the given line to determine the slope of the line that is perpendicular.

y=2x+3.

m = 2

The slope of a line perpendicular to the given line is the opposite reciprocal of  2 / 1

Use −1 / 2 as the slope of new line.

Step 2: Start with the point-slope form. Use the given point and slope of the perpendicular line.

y+5 = −1 / 2  (x – 6)

The graph of y + 5 = −1 / 2 (x – 6) passes through the point (6, -5) and is perpendicular to the graph of y=2x+3.

## Exercise

1. What is the equation of the line parallel to y=3x+5 and through the point (1, 7)?
2. What is the equation of the line parallel to y=4x+3 and through the point (5, 9)?
3. What is the equation of the line that is perpendicular to y=2x+10 and goes through the point (5, 1)?
4. What is the equation of the line parallel to y= 3/4x+1 and through the point (-4, 9)?
5. Determine the equation of a line perpendicular to y=3x−2 at the point (2, 4).
6. What is the equation of the line that passes through (4, 5) and is perpendicular to the graph of y=2x-3?
7. Define parallel lines.
8. Define perpendicular lines.
9. Are the graph of the equations 4y=2x-5 and y=-2x +7 parallel, perpendicular, or neither?
10. Write the equation of a line that is perpendicular to y=−1/2 x+4 and goes through the point (0, 6)?

### What have we learned

• Understand parallel and perpendicular lines
• Write an equation of a line Parallel to given line
• Understand the slope of perpendicular lines
• Understand how to write an equation of a line perpendicular to a given line.

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