For any line, we can define its slope ratio, which tells us how fast the graph climbs or goes down as we look at the line from the left to the right. Use the slider to verify that the higher the slope ratio, the steeper the line will be. Conversely, a flatter line will have a smaller slope ratio.
One way to find the slope ratio of a line is to draw a slope triangle. A slope triangle for any straight line. Slope triangles are right triangles whose hypotenuse connects any two points on the line. The other legs of the triangle are horizontal and vertical segments.
The lengths of its legs are used to write the slope ratio of the line.
$$ \text{slope ratio} = \text{length of vertical line} : \text{length of horizontal line} $$We can also write the ratio in fraction form:
$$ \text{slope ratio} = \frac{\text{length of vertical line}}{\text{length of horizontal line}} $$Example 1: Let's find the slope ratio of the line below using slope triangles.
The first thing to do is pick any two points on the line. In the graph below, we plotted two points that we will be two of the triangle's vertices. To plot the third vertex of the triangle, we need a vertical segment that passes through one of these points and a horizontal segment that passes through the other point. We have two options: we can pick the triangle on top of the line or the triangle below the line.
Say we pick the triangle on top of the line. Now, to get the slope ratio, we need the lengths of the horizontal and vertical legs.
We can see on the graph above that the vertical leg is 4 units long and the horizontal leg is 6 units long. Now we can find the slope ratio of the line by using the lengths of the legs.
$$ \text{slope ratio} = \frac{\text{length of vertical line}}{\text{length of horizontal line}} = \frac{4}{6} $$We can simplify by dividing both numbers by half:
$$ \text{slope ratio} = \frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3} $$So the slope ratio of the line is 4:6, which can be written in fraction form as ⅔.
If we pick different slopes triangles in the same line, do we get different slope ratios for the same line? Let's try it and find out.
Example 2: What happens if we pick slope triangle below versus the one above the line?
Say we have picked the two points shown on the line below. That gives us a slope triangle above the line and a slope triangle below.
The slope ratio from the slope triangle above the line is:
$$ \text{slope ratio} = \frac{\text{vertical line}}{\text{horizontal line}} = \frac{4}{5} $$The slope ratio from the slope triangle below the line is:
$$ \text{slope ratio} = \frac{\text{vertical line}}{\text{horizontal line}} = \frac{4}{5} $$Why do we get the same ratio? Notice that the slope triangles form a rectangle. So both of the vertical segments will have the same length; and both of the horizontal segments will also be of equal length.
In the example above, we drew two different triangles using the same two points on the line and we got the same slope ratio. But what happens if we pick triangles from different points? What happens if the two slope triangles are of different size? Would we get the same slope ratio? Let's try it and find out.
Example 3: What happens if we pick a smaller slope triangle versus a bigger one?
We have a new line below. Let's draw a small slope triangle by picking two points on the line that are close together.
The triangle has a vertical leg that is 1 unit long and a horizontal leg that is 2 units long. The slope ratio is:
$$ \text{slope ratio} = \frac{\text{vertical line}}{\text{horizontal line}} = \frac{1}{2} $$So, using the small triangle, we got that the slope ratio of the line is ½. Let's pick points that are farther away from each other to make a bigger slope triangle.
The bigger triangle has a vertical leg that is 4 units long and a horizontal leg that is 8 units long. The slope ratio is:
$$ \text{slope ratio} = \frac{\text{vertical line}}{\text{horizontal line}} = \frac{4}{8} $$We can simplify the ratio by dividing both numbers by half.
$$ \text{slope ratio} = \frac{4}{8} = \frac{4 \div 2}{8 \div 2} = \frac{1}{2} $$In the examples, we noticed that any given line will have one (and just one) slope ratio that describes it. Different slope triangles might have different dimensions, but the slope ratios we calculate will be the same or equivalent.