Fishtank Learning

8th Grade Math
Unit 7: Pythagorean Theorem and Volume
Lesson 6

Objective

Understand the Pythagorean Theorem as a relationship between the side lengths in a right triangle.

Anchor Problems - Problem 1

Several right triangles are shown below (not drawn to scale). In a right triangle, the two side lengths that form the right angle are called legs, and the side length opposite the right angle is called the hypotenuse.

Right triangles with labeled legs and hypotenuse, varying side lengths.

Use the triangles to investigate the question: What relationship do you see between the measures of the legs and the measure of the hypotenuse?

Guiding Questions

Given the definition of legs and hypotenuse, which side length represents the hypotenuse in each triangle?
What is your strategy for investigating different relationships?
How will you keep track of your work?
If you are stuck, ask your teacher for a clue.
Does the relationship you find work for all of the right triangles shown?
Challenge: A triangle that is similar to triangle A will have side lengths of $$3a$$, $$4a$$, and $$5a$$, for some constant multiplier $$a$$. Will any triangle that is similar to triangle A have the same relationship between the measures of the legs and the measure of the hypotenuse? Use algebra to justify your response.

Notes

The purpose of this Anchor Problem is to allow students time to explore the side lengths of right triangles in search of the relationship that exists between them. After students have enough exploration time, discuss what relationships were discovered and confirmed with repetition (MP.8). Ensure that the discussion ends with the Pythagorean Theorem, $${a^2+b^2=c^2}$$, and confirm this relationship holds for all triangles shown above.
Students may not naturally navigate toward finding the squares of the side lengths, so you can either give them this clue or offer this table for them as a guide:

Triangle	(First leg)$$^2$$	(Second leg)$$^2$$	(Hypotenuse)$$^2$$
A
B
C

Anchor Problems - Problem 2

Use the Pythagorean Theorem to show the relationship between the sides of the right triangles shown below.

Two right triangles with side lengths labeled, one with hypotenuse √40 and legs 6 and 2, the other with hypotenuse √12 and legs √3 and 3.

Anchor Problems - Problem 3

A right triangle has side lengths $$6$$ units, $$4$$ units, and $${\sqrt{20}}$$ units.

Which side length represents the hypotenuse?

Problem Set

View Problem Set

Target Task - Problem 1

Lee tried to use the Pythagorean Theorem on the triangle shown below and found that the relationship did not hold true.

Triangle with sides labeled 7, 5, and √80, showing Pythagorean Theorem application.

Explain why the Pythagorean Theorem did not show a true relationship in Lee’s triangle.

Target Task - Problem 2

Which three measures could be the side lengths of a right triangle? Explain or show your reasoning.

$$5$$ $$2$$ $$7$$ $$\sqrt{24}$$