In this lesson, we'll introduce the concept of slope and then discuss the relationship between the slopes of two perpendicular lines. Knowing this simple relationship will save you time on many math problems!
What is Slope?
The short answer to this question is rise over run, but there is more to it than that. First of all, slope is a term you will see used in algebra because it tells us the angle of the line being graphed. A slope works like a fraction, with the rise, or up/down movement, over the run, or movement from side to side. If a line moved up one unit for every unit it moved to the right, it would have a slope of positive one.
This line moves up 1 unit for every unit it moves right. It has a slope of 1.
For that same line, you could think of it as going down one unit for every unit it moved to the left, and still get the same slope!
Even if we measure the slope in the negative direction, it still has a positive slope!
If the sign of the rise and the run are the same, the line will have a positive slope. However, if the rise and run have opposite signs, then the slope will be negative.
The rise and run of a line with negative slope have opposite signs.
You could say that this line moves to the left while going up, or you could say that it moves to the right while going down. Either way, the signs of the rise and the run are always opposite, so the slope is always negative.
Finding the Slope of a Perpendicular Line
Two lines are perpendicular if they intersect at an angle of exactly 90 degrees. If you know the slope of one of the lines, it's actually quite easy to find the slope of the other. This is because the slopes of any two perpendicular lines are negative reciprocals of one another. That may sound a bit confusing, so let's take a look at an example.
Perpendicular lines have slopes that are negative reciprocals of each other.
Here, we have two lines that are perpendicular to each other. Using the graph, we can see that the blue lines goes up 4 units for every 1 unit it moves to the right. This means it has a slope of 4 over 1. The red line moves down one unit for every 4 units it moves to the right. This means it has a slope of -1 over 4. The fractions 4/1 and -1/4 are negative reciprocals of each other, meaning the numerators and denominators are swapped and the signs are opposite.
This rule holds true for all perpendicular lines and it can save you time on your math homework. For example, if you were asked to find the slope of a line perpendicular to another line that had a slope of -3/4, you wouldn't even need to graph it to find the answer. Using your knowledge of perpendicular lines, you could swap the numerator and denominator of the fraction and change the sign to find the slope of 4/3.
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Your uncle Harry wants you to help him lay out a square pen to hold his baby chickens. He's already laid out one side of the pen and wants you to do the rest. With your trusty compass in hand, you pace out the side already in place and see that it goes 1 foot to the North and 5 feet to the East.
One side of the pen has a slope of 1/5 and the side perpendicular to it has a slope of 5/-1.
Using only your knowledge of perpendicular lines, you pace out 5 feet to the North and 1 foot to the West, or negative East. This means that while the first side of the pen has a slope of 1/5, the perpendicular side has a slope of 5/-1. Harry is going to be mighty impressed with this new pen when you're all done!
Slope is a term you will see used in algebra because it tells us the angle of the line being graphed. A slope works like a fraction, with the rise, or up/down movement, over the run, or movement from side to side.
Two lines are perpendicular if they intersect at an angle of exactly 90 degrees. The slopes of any two perpendicular lines are negative reciprocals of one another, meaning the numerators and denominators are swapped and the signs are opposite. For example, if a line has a slope of 4/3, a line perpendicular to it will have a slope of -3/4. This relationship is true for all perpendicular lines.
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