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Proving Slope
using Similar Triangles
1
Launch
The Essential Question
“How is the slope of a line related
to similar triangles”
Agenda
2
Launch
Let’s take a look at the formal definition for similar triangles:
Similar Triangles: If two angles of a triangle have measures
equal to the measures of two angles of another triangle,
then the triangles are similar. Corresponding sides
of similar polygons are in proportion, and corresponding
angles of similar polygons have the same measure.
Agenda
3
Launch
The Second Piece to our Puzzle!
What is slope?
Slope: The slope of a line is a number that measures its
"steepness", usually denoted by the letter m. It is the
change in y for a unit change in x along the line.
How can you calculate the slope between two distinct points?
m
y2  y1
x2  x1
Let’s take a look at an example,
Find the slope of the line segment joining the points ( 1, - 4 ) and ( - 4, 2 ).
y2  y1 2  4
6
m


x2  x1 4  1
5
Agenda
4
Explore
In the graph below, complete the following steps:
• Choose and plot two points above the line given. Label the points A and B.
• Choose and plot two points below the line given. Label the points C and D.
• Choose and plot any two points on the line given. Label the points M and N.
Agenda
5
Explore
Let’s take a look at what points I chose:
Agenda
6
Explore
1. Calculate the slope of the line given using points M and N.
2. What do you notice that is similar about the slope and the ratios of the side
length for the legs of each triangle for points A, B, C, and D?
Agenda
7
Explore
For the points I chose, I had…
1. Calculate the slope of the line given using points M and N.
y2  y1 4  1 5 1
m



x2  x1 6  4 10 2
2. What do you notice that is similar about the slope and the ratios of the side
length for the legs of each triangle for points A, B, C, and D?
rise 5.5 3.5 4 5.5 1
m


 

run 11
7
8 11 2
Agenda
8
Practice
What is my slope?
For each problem:
• Identify three distinct points (try different points than you have already used).
• Create three similar right triangles using the two of the three distinct points for
each hypotenuse. Label the point for each right angle.
• Measure the change in x and y for each triangle.
• Write the change of y over the change of x to represent the slope m for the line.
Agenda
9
Practice
What is my slope?
m3
2
m
3
1
m
4
Agenda
10
Practice
m3
2
m
3
1
m
4
Agenda
11
Summary
What can you conclude about the
right triangles?
What is the same?
What is different?
Agenda
12
Assessment
Exit Slip: On the line given, choose one distinct point B
between given points A and C. Using three similar triangles
show that the slope m is the same between A, B, and C.
Agenda
13