Topics
9. Dilations & Similarity
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United States of America
Grade 8

9.05 Slope and similar triangles

Lesson
Practice

Introduction

We've learned that in similar shapes, all corresponding sides are in the same ratio and all corresponding angles are equal. We have also seen how this holds on the coordinate plane.

Let's look at how similar triangles relate to slope in the coordinate plane.

The slope formula and similar triangles

We find the slope by using the formula:

m=\dfrac{\text{rise}}{\text{run}} \text{, or}

m=\dfrac{y_2-y_1}{x_2-x_1}

Exploration

Let's tie in some previous understandings we developed about triangles, slopes, and angles.

Use the applet below to explore the connection between similar triangles and slope.

  1. Notice the angle measures of each triangle are labeled in the bottom left and top right corners of the applet.

  2. Click and drag the blue points L_1 and L_2 to change the slope of the line.

  3. As you change the slope what changes? What doesn't change?

Loading interactive...

From the applet we can see that, no matter how we change the slope, the measures in all three angles are the same for both triangles. We can also see that one of the angles is always a right angle in both triangles.

Since all the triangles have the same sized angles, this is one of our definitions of similarity. So all triangles formed (in this manner) from a line with the same slope are similar.

Examples

Example 1

Consider the points A, B, and C.

-1
1
2
3
4
5
6
7
x
1
2
3
4
5
6
7
8
9
10
11
12
13
y
a

Complete the directions that explain how to move from point A to point B:

From A, move ⬚ units up, and ⬚ units to the right.

Worked Solution
Create a strategy

We need to find the difference in the y-values (i.e. rise) between points A and B , and the difference between their x-values (i.e. run).

Apply the idea
\displaystyle \text{rise}\displaystyle =\displaystyle 8-0Subtract the y-coordinates of points A and B
\displaystyle =\displaystyle 8\text{ units}Evaluate
\displaystyle \text{run}\displaystyle =\displaystyle 4-0Subtract the x-coordinates of points A and B
\displaystyle =\displaystyle 4\text{ units}Evaluate

From A, move 8 units up, and 4 units to the right.

b

Express the direction of the movement in the previous question as a simplified ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a\text{:}b.

Worked Solution
Create a strategy

Use the slope formula: m=\dfrac{\text{rise}}{\text{run}}, then convert to ratio.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac84Substitute the value of the rise and run
\displaystyle =\displaystyle \dfrac21Evaluate
\displaystyle =\displaystyle 2\text{:}1Express as a ratio
c

Complete the directions that explain how to move from point A to point C.

From A, move ⬚ units up, and ⬚ units to the right.

Worked Solution
Create a strategy

We need to find the difference in the y-values (i.e. rise) between points A and C , and the difference between their x-values (i.e. run).

Apply the idea
\displaystyle \text{rise}\displaystyle =\displaystyle 12-0Subtract the y-coordinates of points A and C
\displaystyle =\displaystyle 12\text{ units}Evaluate
\displaystyle \text{run}\displaystyle =\displaystyle 6-0Subtract the x-coordinates of points A and C
\displaystyle =\displaystyle 6\text{ units}Evaluate

From A, move 12 units up, and 6 units to the right.

d

Express the direction of the movement in the previous question as a simplified ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a\text{:}b.

Worked Solution
Create a strategy

Use the slope formula: m=\dfrac{\text{rise}}{\text{run}}, then convert to ratio.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac{12}{6}Substitute the value of the rise and run
\displaystyle =\displaystyle \dfrac21Evaluate
\displaystyle =\displaystyle 2\text{:}1Express as a ratio

Example 2

Which of the triangles listed below are similar to \triangle{FDB}?

-8
-6
-4
-2
2
4
6
x
-12
-10
-8
-6
-4
-2
2
4
6
8
10
y
A
\triangle{EFA}
B
\triangle{CBA}
C
\triangle{AEB}
D
\triangle{CFB}
Worked Solution
Create a strategy

Choose that triangle that also has a right triangle and has a side with the same slope as \overline{FD}.

Apply the idea

The correct option is A: \triangle{EFA}, because it is a right triangle and \overline{AF} has the same slope as \overline{FD}.

Idea summary

We find the slope of a line by using the formula:

\displaystyle m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}
\bm{(x_1,y_1)}
are the coordinates of the lower points
\bm{(x_2,y_2)}
are the coordinates of the upper points

All triangles formed from a line with the same slope are similar.

Outcomes

8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

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