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)?

Concept Map 

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.