PRINT VERSION - Pythagorean Theorem to Find Distance

Given points on a coordinate plane, students are able to construct right triangles and use the Pythagorean theorem to find missing distances. Students apply their knowledge of the area of squares to make connections between the side lengths of right triangles and their squares in order to find unknown side lengths.

Goals and Standards

---

Colorado Academic Standards:

• 8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Mathematical Practice Standards:

• MP 4 Model with mathematics 
• MP 5 Use appropriate tools strategically (coordinate plane)
• MP 7 Look for and make use of structure 

Learning Goal:

• Construct right triangles, rectangles and squares on the coordinate plane 
• Find the distance between two points on the coordinate plane 
• Apply Pythagorean theorem to find missing side lengths of right triangles

Vocabulary

• distance
• hypotenuse
• leg
• Pythagorean theorem
• right triangle
• square root
• squared

Next Section:

Progression of Skills (outline)

Before the lesson:

• How to find the area of a square
• Using square roots and squared numbers as inverse operations
• Graphing ordered pairs on the coordinate plane
• Identifying a right triangle and its hypotenuse
• Basic understanding of the Pythagorean Theorem

During this toolkit lesson:

• Creating squares and right triangles from points on the coordinate plane
• Using the Pythagorean Theorem to find unknown side lengths
• Calculating square roots as the inverse of a value squared

After the lesson:

• Using Pythagorean Theorem to identify right triangles in a variety of real world scenarios to solve for unknown side lengths

Lesson Session 1

Session 1 student worksheet

Directions for Intervention Activity:

1. Display the ordered pairs below. Provide students with graph paper or Geogebra digital coordinate plane

Graph of ordered pairs Point A (1,1), Point B (5,1), and Point C (3,7)

2. Ask students to find the distance between Point A and Point B (#1 on session 1 worksheet)
3. Ask students to think about how to find the distance between Point A and Point C (#2 on session 1 worksheet)

Questions to pose to students:

• "Why is it more difficult to find the exact distance between Point A and Point C?"
• "How do we use the coordinate grid to measure distances?"
• "What can we do if the distance we are measuring does not create a vertical or horizontal line?"

4. Remind students of the Pythagorean Theorem formula, a2+b2=c2 . Redirect students to vocabulary if needed. Notice the difference between the capital A, B, and C that stand for the points whereas the lower case a, b, and c stand for the lengths of the sides of a triangle.
5. Ask students to construct a right triangle with Points A, C, and Point D. (#3 on session 1 worksheet)

Questions to pose to students:

• "What shape is needed to use the Pythagorean theorem?"
• "Is there a way you can create a right triangle using Points A and Points C and a new point, Point D?"
• "What does AC represent in the formula? How can you identify the hypotenuse in a right triangle?"
• "Does it matter if you place Point D at (1,7) or (3,1)?"

6. Ask, "Which side in the right triangle ACD is the hypotenuse? Which sides are the legs?" hypotenuse = AC, legs = AD and CD
7. Ask students to measure and label sides AD and DC. AD = 2 CD = 6

The rest of these steps help students work through #4 on the session 1 worksheet:

8. Guide students to substitute a and b in the formula with side lengths of AD and DC.
9. **What if AD is along the side, not on the bottom?**
10. Solve for c in the formula. Round your answer to the nearest tenth.
    1. Substitute the legs for a and b in the formula and simplify: a2+b2=c2
    2. Use this opportunity to remind students that the inverse of squaring a number is to find the square root. Discuss that not all numbers are perfect squares, that the square root of c may not be a whole number.

Progress Monitoring:

• Frequent checks for understanding through guiding questions that lead to student discussion
• Check student comprehension on post-assessment

Directions for Session 1 Closure:

1. Display ordered pairs below:

Graph of ordered pairs Point A (-1,1) and Point B (2,-4)

2. Ask students to construct a right triangle using Point A and Point B to find the distance between Point A and Point B using the Pythagorean Theorem. Round your answer to the nearest tenth. (#5 on session 1 worksheet)

Post-assessment:

• A running track runs from point (1, 2) to point (5, 5). Another running track goes from point (5, 5) to point (8, 9). What is the straight-line distance between the starting point (1, 2) and the ending point (8, 9)? Round your answer to the nearest whole number. Use graph paper if needed. (#6 on session 1 worksheet)

Questions to pose to students:

• "How can we solve this problem without graph paper?"
• "Can you find the length of the two legs by finding the distance between the points?"

Real World Applications and Extensions for Session 1

• A spider is positioned at point I (-3, 5) and moves to point J (2, -1) to build its web. Calculate the distance the spider travels directly from point I to point J. Use graph paper if needed. (#7 on session 1 worksheet)

Lesson Session 2

Session 2 student worksheet

Directions for Intervention Activity:

1. Give students the prompt, "Plot the points A (5,3), B (-1,1), and C (2,-3) in the coordinate plane and find the lengths of the three segments connecting the points." Provide students with graph paper or Geogebra digital coordinate plane (#1 on session 2 worksheet)

Questions to pose to students:

• "What shape is created when you connect the three points? What kind of triangle is it?"
• "Can Pythagorean theorem be applied to any triangle?"
• "If the points connect to make a triangle without a right angle, what could you do to create other shapes that do have right angles?"

Students may complete this task individually, in small groups, or as a whole class discussion.

2. Monitor students as they plot the points and discuss how to approach the task. If students begin using the Pythagorean Theorem with Triangle ABC, remind them that the Pythagorean Theorem only works with right triangles and Triangle ABC is not a right triangle.
3. Guide students to use the perpendicular grids on the coordinate plane to construct a rectangle around the three ordered pairs. From this rectangle, students will be able to construct three right triangles with AB, BC, and CA as the hypotenuses. (#2 on session 2 worksheet)

Questions to pose to students:

• "Can you construct any right triangles, rectangles, or squares?"
• "Can you dissect these shapes into a shape we could use Pythagorean Theorem on?"

Student work example (drawings may vary - students should create three right triangles with AB, BC, and AC as the hypotenuses)

Graph of student work example

Progress Monitoring:

• Frequent checks for understanding through guiding questions that lead to student discussion
• Check student comprehension on post-assessment

• Carlos has a rectangular garden that measures 10 feet by 24 feet. He needs to walk from one corner of the garden to the opposite corner to water his plants. What is the shortest distance he has to walk? (#4 on session 2 worksheet)

• Güner, N. (2018). How to teach the Pythagorean Theorem: an analysis of lesson plans. Journal of Faculty of Educational Sciences, 51(1), 119–141. https://doi.org/10.30964/auebfd.405041