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 Lesson Plan: Slope and Similar Triangles

 

Lesson Summary

In this lesson, students will explore how the concept of similar triangles can be applied to understand why the slope between any two distinct points on a non-vertical line remains constant.

Through interactive activities, including drawing right triangles on a coordinate plane and analyzing their properties, students will visually and analytically verify the consistency of slope. The lesson incorporates multimedia resources to reinforce key concepts and provides opportunities for students to apply their knowledge through problem-solving exercises.

Lesson Objectives

  • Students will understand the concept of slope using similar triangles.
  • Students will be able to explain why the slope is the same between any two distinct points on a non-vertical line.

Standards

  • CCSS.MATH.CONTENT.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.

Prerequisite Skills

  • Understanding of Right Triangles: Ability to identify and define right triangles, including recognizing the legs and hypotenuse.
  • Familiarity with Similar Triangles: Knowledge of the criteria that determine triangle similarity, such as corresponding angles and proportional side lengths.
  • Coordinate Plane Proficiency: Experience with plotting points, drawing shapes, and interpreting coordinates on a Cartesian plane.
  • Basic Ratio and Proportion Concepts: Understanding how to form and manipulate ratios and proportions, which are fundamental in comparing side lengths of similar triangles.

Key Vocabulary

  • Slope: A measure of the steepness or incline of a line, calculated as the ratio of the rise to the run between two points on the line.
  • Rise: The vertical change between two points on a line.
  • Run: The horizontal change between two points on a line.
  • Ratio: A comparison of two quantities by division.
  • Similar Triangles: Triangles that have the same shape but may differ in size, characterized by equal corresponding angles and proportional corresponding side lengths.
  • Right Triangle: A triangle that has one 90-degree angle.
  • Hypotenuse: The longest side of a right triangle, opposite the right angle.
  • Legs of a Right Triangle: The two sides that form the right angle in a right triangle.
  • Proportional: Having a constant ratio between corresponding quantities.
  • Coordinate Plane: A two-dimensional plane formed by the intersection of a vertical y-axis and a horizontal x-axis, used for plotting points, lines, and curves.

Multimedia Resources

 

 

Warm Up Activities

Use one or more of these activities.

Activity 1: Review of Right Triangles

Activity 2: Review of Similar Triangles

  • Begin by presenting two pairs of triangles, one pair being similar and the other not.
  • Ask students to identify corresponding angles and determine whether the triangles are similar using the Angle-Angle (AA) similarity criterion.
  • Have students complete a short matching exercise where they pair similar triangles based on given angle measures and side ratios.
  • Discuss why similarity is important in understanding slope and proportional relationships.

Activity 3: Review of Proportions

  • Provide students with simple proportion problems, such as solving for missing values in equivalent ratios (e.g., \( \frac{3}{4} \) = \( \frac{x}{8} \)).
  • Introduce a real-world example, such as scaling a recipe or comparing map distances to actual distances.
  • Have students complete a quick "proportions scavenger hunt" by finding examples of proportional relationships in classroom objects (e.g., desk widths vs. lengths).
  • Conclude with a discussion on how proportions relate to slope and triangle similarity.

 

 

Teach

Introduction to the Lesson

In this lesson, students will explore the relationship between slope and similar triangles. They will learn how to use the properties of proportional side lengths in right triangles to explain why the slope between any two points on a line remains constant.

Students will work with the coordinate plane to draw and analyze triangles formed by two points on a line. They will apply their understanding of ratios and proportions to determine how the vertical change (rise) and horizontal change (run) create a consistent ratio, which defines the slope.

By the end of this lesson, students will be able to explain why similar triangles always produce the same slope and how this concept is fundamental to understanding linear relationships.

Show the following video, which shows how similar right triangles can be used to explore slope:

https://www.media4math.com/library/75382/asset-preview 

 

Example 1: Finding Slope Using Right Triangles

Given two points \( A(2,3) \) and \( B(6,7) \), determine the slope by constructing a right triangle and using similar triangles.

  1. Plot the points: Draw a coordinate plane and place points at \( A(2,3) \) and \( B(6,7) \).
  2. Create a right triangle:
    • Draw a horizontal line from \( A \) to directly under \( B \).
    • Draw a vertical line from \( B \) to meet the horizontal line at a right angle.
  3. Identify the rise and run:
    • Rise: \( 7 - 3 = 4 \)
    • Run: \( 6 - 2 = 4 \)
  4. Calculate the slope: \[ m = \frac{\text{rise}}{\text{run}} = \frac{4}{4} = 1 \]

 

Slope

 

Example 2: Proving Slope is the Same Using Similar Triangles

Show that the slope remains the same for two different sets of points on the line passing through \( (2,3) \) and \( (6,9) \).

  1. Find the slope for the full segment: \[ m = \frac{9-3}{6-2} = \frac{6}{4} = \frac{3}{2} \]
  2. Choose another point on the line, such as \( (4,6) \), and form a smaller right triangle.
  3. Calculate the slope using the smaller triangle: \[ m = \frac{6-3}{4-2} = \frac{3}{2} \]
  4. Compare the two right triangles and explain how their slopes match.
  5. Corresponding sides of the two right triangles are proportional.

 

Slope

 

Example 3: Determining an Unknown Coordinate Given a Slope

The points \( A(3,5) \) and \( B(x,9) \) lie on a line with a slope of \( \frac{2}{3} \). Find \( x \).

 

Slope

 

  1. Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  2. Substitute values: \[ \frac{9 - 5}{x - 3} = \frac{2}{3} \]
  3. Solve the proportion: \[ 4 = \frac{2}{3} (x - 3) \]
  4. Multiply both sides by 3: \[ 12 = 2(x - 3) \]
  5. Divide by 2 and solve for \( x \): \[ x = 9 \]

Example 4: Solving for Ramp Height Using a Proportion

A wheelchair ramp is 12 feet long and has a slope of \( \frac{1}{4} \). Find the height of the ramp.

  1. Use the slope formula: \[ m = \frac{\text{rise}}{\text{run}} \]
  2. Substitute the given values: \[ \frac{h}{12} = \frac{1}{4} \]
  3. Cross multiply: \[ h = \frac{12 \times 1}{4} = \frac{12}{4} = 3 \]
  4. Conclude that the height of the ramp is 3 feet.

Example 5: Estimating the Height of a Tree Using a Laser Pointer

A student stands 15 feet from a tree and points a laser at a 40° angle to the top. The laser forms a line with a slope of \( \frac{3}{5} \). Estimate the height of the tree.

  1. Use the slope formula: \[ m = \frac{\text{rise}}{\text{run}} \]
  2. Substitute the given values: \[ \frac{h}{15} = \frac{3}{5} \]
  3. Cross multiply: \[ h = \frac{15 \times 3}{5} = \frac{45}{5} = 9 \]
  4. Conclude that the estimated height of the tree is 9 feet.

 

 

Review

In this lesson, you learned how to determine the slope of a line and saw the connection between similar right triangles and slope. By constructing right triangles on a coordinate plane, we demonstrated that slope remains constant along a straight line. We also explored how slope is calculated using the rise (vertical change) and run (horizontal change) between two points. Finally, we applied these concepts to real-world scenarios, including ramps, hiking trails, and estimating the height of objects using indirect measurement.

  • Use these clip art images to see if students can determine if the two right triangles are similar.

https://www.media4math.com/library/75380/asset-preview

  •  For each of the lines shown, have students calculate the slope. Have them use the underlying grid to make their calculations. Then have them compare the slopes.

https://www.media4math.com/library/75381/asset-preview

Key Vocabulary

  • Slope: The measure of steepness of a line, found by the ratio of vertical change (rise) to horizontal change (run).
  • Rise: The vertical difference between two points on a line.
  • Run: The horizontal difference between two points on a line.
  • Similar Triangles: Triangles that have the same shape but may have different sizes. They have corresponding angles that are equal and corresponding sides that are proportional.
  • Right Triangle: A triangle that has one 90-degree angle.
  • Proportional: Two ratios or fractions that are equal to each other.
  • Coordinate Plane: A two-dimensional grid defined by an x-axis (horizontal) and y-axis (vertical), used to graph points and lines.

Example 1: Finding an Unknown Coordinate with a Given Slope

The points \( A(1,2) \) and \( B(x,8) \) lie on a line with a slope of \( \frac{3}{4} \). Find \( x \).

  1. Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  2. Substitute the given values: \[ \frac{8 - 2}{x - 1} = \frac{3}{4} \]
  3. Solve for \( x \): \[ 6 = \frac{3}{4} (x - 1) \] Multiply both sides by 4: \[ 24 = 3(x - 1) \] Divide by 3: \[ x - 1 = 8 \] Add 1 to both sides: \[ x = 9 \]

Example 2: Real-World Application – Designing a Staircase

A staircase must have a vertical rise of 9 feet. The building code requires the slope of the staircase to be \( \frac{3}{5} \). Find the total horizontal length of the staircase.

  1. Use the slope formula: \[ m = \frac{\text{rise}}{\text{run}} \]
  2. Substitute the given values: \[ \frac{9}{x} = \frac{3}{5} \]
  3. Cross multiply: \[ 3x = 9 \times 5 \] \[ 3x = 45 \] \[ x = \frac{45}{3} = 15 \]
  4. The total horizontal length of the staircase must be 15 feet.

Example 3: Estimating the Height of a Billboard

A person is standing 20 feet away from a billboard and shines a laser pointer at a 50° angle to the top of the billboard. The laser follows a line with a slope of \( \frac{5}{4} \). Estimate the height of the billboard.

  1. Use the slope formula: \[ m = \frac{\text{rise}}{\text{run}} \]
  2. Substitute the given values: \[ \frac{h}{20} = \frac{5}{4} \]
  3. Cross multiply: \[ h = \frac{20 \times 5}{4} = 25 \]
  4. The estimated height of the billboard is 25 feet.

 

Quiz 

Test your understanding of slope and similar triangles by answering the following questions.

  1. Find the slope of the line passing through the points \( (2,3) \) and \( (6,7) \).
  2. Are the triangles formed by the points \( (1,2) \), \( (5,6) \), and \( (9,10) \) similar? Why?
  3. The points \( A(3,4) \) and \( B(x,10) \) lie on a line with a slope of \( \frac{3}{2} \). Find \( x \).
  4. A wheelchair ramp has a slope of \( \frac{1}{6} \) and a run of 18 feet. Find the height.
  5. A tree casts a shadow of 20 feet while a 5-foot pole casts a shadow of 4 feet. Are the triangles similar? Explain.
  6. A road has a vertical rise of 15 feet over a horizontal run of 75 feet. What is the slope of the road?
  7. The points \( (2,5) \), \( (4,9) \), and \( (6,13) \) lie on the same line. Show that the slope is the same for all segments.
  8. A student stands 12 feet from a lamppost and shines a laser at the top. The laser follows a line with a slope of \( \frac{5}{6} \). Find the height of the lamppost.
  9. Two ramps are designed with the same slope of \( \frac{2}{5} \). If one ramp is 10 feet long, how long is the second ramp if it has twice the height?
  10. A hiking trail gains 200 feet in elevation over a horizontal distance of 800 feet. What is the slope of the trail? Express your answer as a simplified fraction.

Answer Key

  1. \[ m = \frac{7-3}{6-2} = \frac{4}{4} = 1 \]
  2. Yes, because they have the same slope, meaning they maintain proportional rise and run.
  3. \[ \frac{10 - 4}{x - 3} = \frac{3}{2} \] Solving for \( x \): \[ 6 = \frac{3}{2} (x - 3) \] Multiply by 2: \[ 12 = 3(x - 3) \] Solve for \( x \): \[ x = 7 \]
  4. \[ \frac{h}{18} = \frac{1}{6} \] Cross multiply: \[ h = \frac{18 \times 1}{6} = 3 \] The height is 3 feet.
  5. Yes, because both create right triangles and their corresponding sides are proportional: \[ \frac{5}{4} = \frac{h}{20} \]
  6. \[ m = \frac{15}{75} = \frac{1}{5} \]
  7. - Between \( (2,5) \) and \( (4,9) \): \[ m = \frac{9-5}{4-2} = \frac{4}{2} = 2 \] - Between \( (4,9) \) and \( (6,13) \): \[ m = \frac{13-9}{6-4} = \frac{4}{2} = 2 \] Since both slopes are equal, the points lie on the same line.
  8. \[ \frac{h}{12} = \frac{5}{6} \] Cross multiply: \[ h = \frac{12 \times 5}{6} = 10 \] The height of the lamppost is 10 feet.
  9. Let \( h \) be the height of the first ramp: \[ \frac{h}{10} = \frac{2}{5} \] Solve for \( h \): \[ h = \frac{10 \times 2}{5} = 4 \] The second ramp has twice the height, so \( 2 \times 4 = 8 \). Use the slope formula: \[ \frac{8}{x} = \frac{2}{5} \] Solve for \( x \): \[ x = \frac{8 \times 5}{2} = 20 \] The second ramp is 20 feet long.
  10. \[ m = \frac{200}{800} = \frac{1}{4} \]