ERPB - Similar Triangles and Slope Lesson

Similar Triangles and Slope

In earlier lessons, you learned about proportions and how to set them up and solve them. Then you learned to use scale drawings and scale models to solve problems. Next, you learned that similar figures are figures that have the same shape - they have congruent corresponding angles and proportional corresponding sides. You learned to set up a proportion to find the missing side lengths. When two polygons are similar, the symbol ~ is used. When you know that polygons are similar, this allows you to find unknown lengths and angle measures. Finally, you learned that the ratio of the lengths of corresponding sides in similar figures is sometimes called their scale factor.

In this lesson, you will put all of that together and use proportional reasoning to explain why the slope, m, is the same between any two distinct points. You will learn to demonstrate a conceptual understanding of slope and use graphical reasoning to represent proportional relationships.

Remember, slope is the steepness of a line. It is the ratio of the vertical change to the horizontal change in any two points on the line. Slope is better known as Rise over Run, “Rise/Run”.  

Image of a graph of a line that runs through the point two, three and the point six, five. It runs 4 and rises 2 to get to the next point.

LaTeX: slope=\frac{rise}{run}=\frac{2}{4}=\frac{1}{2}slope=riserun=24=12

  • What mathematical operation could you have used on the y-values to discover rise?
  • What mathematical operation could you have used on the x-values to discover run?

Subtraction

Earlier we said that slope is the ratio of the vertical change to the horizontal change. We could write this ratio as (change in y values)/(change in x values). We just discovered that we find that change using subtraction. Now we can write the formula for slope.

If a nonvertical line passes through the points, (x1, y1) and (x2, y2), the slope, m, is LaTeX: \frac{y_2-y_1}{x_2-x_1}y2−y1x2−x1.

Graph showing the change in y over the change in x.

Investigate

We can test to see if this works by using the graphs below.

Find the slope of the following graphs using the given points and the formula. Then check your work using rise/run on the graph.

1. Graph for activity with a positive slope.

2. Graph for activity with a negative slope.

Check your answers.

Using points:

Graph 1: LaTeX: \frac{(6-3)}{(4-2)}=\frac{3}{2}=1\frac{1}{2}(6−3)(4−2)=32=112    Graph 2: LaTeX: \frac{(7-5)}{(2-4)}=\frac{2}{-2}=-1(7−5)(2−4)=2−2=−1

Using the graph:

Graph 1: LaTeX: \frac{rise}{run}=\frac{3}{2}=1\frac{1}{2}riserun=32=112  Graph 2: LaTeX: \frac{rise}{run}=\frac{2}{-2}=-1riserun=2−2=−1

Image showing graphs with different slopes.

What else can we discover about slope? Use the graph below to answer the questions.

Graph of a line with different point on it.

  1. Pick two points on the line to find the slope.
  2. Pick two different points on the line to find slope.
  3. Do you notice a pattern?
  • Pick two points on the line to find the slope. (-4, -5) and (-2, -2) = LaTeX: \frac{(-2--5)}{(-2--4)}=\frac{3}{2}=1\frac{1}{2}(−2−−5)(−2−−4)=32=112
  • Pick two different points on the line to find slope. (0,1) and (2,4) = LaTeX: \frac{(4-1)}{(2-0)}=\frac{3}{2}=1\frac{1}{2}(4−1)(2−0)=32=112
  • Do you notice a pattern? The slope is the same or constant! That means no matter what two points you choose on a non-vertical line, you can find the slope.

Now let’s look at similar triangles to prove the slope is constant.

Remember, for an object to be similar, it must have corresponding angles that are congruent and corresponding sides that are proportional (the ratio of any two corresponding sides will be equal).

Understanding Slope with Similar Triangles

Investigate

Investigate

The data shown in the graph below reflects average wages earned by assembly line workers across the nation.

Graph of Average wages earned by assembly line workers

Answer the following questions using the graph and check your work.

  1. What hourly rate is indicated by the graph? Explain how you determined your answer.
  2. What is the ratio of the height to the base of the small, medium, and large triangles?
  3. Are these ratios proportional?
  4. The slope of a line is found by forming the ratio of the change in y to the change in x between any two points on the line. What is the slope of the line formed by the data points in the graph above? Explain how you know.

Let’s check our answers.

  1. What hourly rate is indicated by the graph? Explain how you determined your answer. The average assembly line worker is paid $15 per hour. The hour is the independent variable and the wages earned is the dependent variable. As the independent variable increases by one, the dependent variable increases by fifteen.
  2. What is the ratio of the height to the base of the small, medium, and large triangles?
    • LaTeX: small=\frac{15dollars}{1hour};medium=\frac{30dollars}{2hours};large=\frac{60dollars}{4hours}small=15dollars1hour;medium=30dollars2hours;large=60dollars4hours
  3. Are these ratios proportional? Yes, they are equivalent which means they are proportional. (you can reduce the medium and large to be 15 dollars/1hour)
  1. The slope of a line is found by forming the ratio of the change in y to the change in x between any two points on the line. What is the slope of the line formed by the data points in the graph above? Explain how you know.
    • LaTeX: small=\frac{15dollars}{1hour};medium=\frac{30dollars}{2hours};large=\frac{60dollars}{4hours}small=15dollars1hour;medium=30dollars2hours;large=60dollars4hours

All of these simplify to a unit rate of $15.00/1hour.

This means the slope is 15.

Similar Triangles and Slope Practice

 Download your practice by CLICKING HERE. Links to an external site. Once you have completed your practice, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your practice key.Links to an external site.

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