3.09 Slopes of lines and similar triangles
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.
We find the slope by using the formula:
$m=\frac{rise}{run}$m=riserun
or
$m=\frac{y_2-y_1}{x_2-x_1}$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?
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.
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?
$\triangle EFA$△EFA
A$\triangle CBA$△CBA
B$\triangle AEB$△AEB
C$\triangle CFB$△CFB
D