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
- What is the equation of the line parallel to y=3x+5 and through the point (1, 7)?
- What is the equation of the line parallel to y=4x+3 and through the point (5, 9)?
- What is the equation of the line that is perpendicular to y=2x+10 and goes through the point (5, 1)?
- What is the equation of the line parallel to y= 3/4x+1 and through the point (-4, 9)?
- Determine the equation of a line perpendicular to y=3x−2 at the point (2, 4).
- What is the equation of the line that passes through (4, 5) and is perpendicular to the graph of y=2x-3?
- Define parallel lines.
- Define perpendicular lines.
- Are the graph of the equations 4y=2x-5 and y=-2x +7 parallel, perpendicular, or neither?
- 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.
Related topics
Square 1 to 20 : Chart, Table, Perfect Squares and Examples
Square 1 to 20 When you multiply a number by itself, the result is called a square. And when you’re preparing for exams, you need to have a foundation for algebra and quick mental math because you get a really short time to do your exam. Therefore, learning the squares from one to twenty is […]
Square 1 to 40 : Table, Perfect Squares, Chart and Examples
Square 1 to 40 When you multiply a number by itself, the resulting number is a square, and if you are someone who is either appearing in a competitive exam or just wants to do well in math in school, knowing square 1 to 40 is a really important skill. But manually multiplying every time, […]
Square Root : Definition, Formula, Methods and Types Explained
Square Root Square roots are one of those seemingly daunting maths topics that appear in many different situations, from algebra to geometry. Yet the concepts behind them aren’t as hard to grasp. It makes handling numbers far easier if you know the concept well. Let us understand how to find the square roots of a number […]
Cubes 1 to 20 : Chart, Table, Memory Tricks and Examples
Most students don’t struggle much with smaller cubes like 2³ or 3³. Those usually come quickly. The hesitation starts with numbers like 11³ or 17³. Or when someone suddenly asks, what is 20 cubed? That pause is not a memory problem. It’s about the lack of proper understanding and hence confidence. Naturally, learning cubes 1 […]
Other topics
Comments: