4.05 Slopes of lines and similar triangles

Lesson

Practice

Lesson

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.

Remember!

We find the slope by using the formula:

$m=\frac{rise}{run}$m=riserun

$m=\frac{y_2-y_1}{x_2-x_1}$m=y2−y1x2−x1

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

Exploration

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

Notice the angle measures of each triangle are labeled in the bottom left and top right corners of the applet.
Click and drag the blue points $L_1$L1 and $L_2$L2 to change the slope of the line.
As you change the slope what changes? What doesn't change?

Created with Geogebra

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.

Practice questions

Question 1

Consider the points $A$A, $B$B and $C$C.

Loading Graph...

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

From $A$A, move $\editable{}$ units up and $\editable{}$ units to the right.
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 $a:b$a:b.

$\editable{}:\editable{}$:
Complete the directions that explain how to move from point $A$A to point $C$C.

From $A$A, move $\editable{}$ units up and $\editable{}$ units to the right.
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 $a:b$a:b.

$\editable{}:\editable{}$:

Question 2

Which of the triangles listed below are similar to $\triangle FDB$△FDB?

Loading Graph...

$\triangle EFA$△EFA
A
$\triangle CBA$△CBA
B
$\triangle AEB$△AEB
C
$\triangle CFB$△CFB
D

Outcomes

NC.8.F.4

Analyze functions that model linear relationships. • Understand that a linear relationship can be generalized by y = mx + b. • Write an equation in slope-intercept form to model a linear relationship by determining the rate of change and the initial value, given at least two (x, y) values or a graph. • Construct a graph of a linear relationship given an equation in slope-intercept form. • Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of the slope and y-intercept of its graph or a table of values.