unit 7 geometry homework 6 the pythagorean theorem answer key

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

Pythagorean Theorem and Volume

Lesson 6 of 16

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

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

Common Core Standards

Core standards.

The core standards covered in this lesson

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Understand the relationship between the legs and the hypotenuse of right triangles, named the Pythagorean Theorem :  $${a^2+b^2=c^2}$$ .
• Use the Pythagorean Theorem to verify the relationship between the legs and hypotenuse of right triangles.
• Understand that the hypotenuse of a right triangle is the longest side of the triangle located across from the right angle.

Suggestions for teachers to help them teach this lesson

In this lesson, students are introduced to the famous relationship that exists between the side lengths of right triangles. In the next lesson, students will investigate informal proofs of the Pythagorean Theorem, followed by lessons where students will apply the theorem in various problems.

Lesson Materials

• Graph Paper (1 sheet per student)

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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

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. 

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

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

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

Which side length represents the hypotenuse?

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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

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

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

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

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include problems where students use given side lengths or identify side lengths from a group for a right triangle and verify the relationship using the Pythagorean Theorem.
• Open Up Resources Grade 8 Unit 8 Practice Problems — Lesson 6 #2-3
• Open Middle Pythagorean Theorem 2 — Challenge
• Open Middle Pythagorean Theorem 1

Topic A: Irrational Numbers and Square Roots

Define, evaluate, and estimate square roots.

Understand that some numbers, including $${\sqrt{2}}$$ , are irrational. Approximate the value of irrational numbers.

8.NS.A.1 8.NS.A.2

Locate irrational values approximately on a number line. Compare values of irrational numbers.

Represent rational numbers as decimal expansions.

Represent decimal expansions as rational numbers in fraction form.

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Topic B: Understanding and Applying the Pythagorean Theorem

Understand a proof of the Pythagorean Theorem.

Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.

Find missing side lengths involving right triangles and apply to area and perimeter problems.

Solve real-world and mathematical problems using the Pythagorean Theorem (Part I).

Solve real-world and mathematical problems using the Pythagorean Theorem (Part II).

Find the distance between points in the coordinate plane using the Pythagorean Theorem.

Topic C: Volume and Cube Roots

Define and evaluate cube roots. Solve equations in the form  $${x^2=p}$$  and $${x^3=p}$$ .

8.EE.A.2 8.NS.A.2

Solve real-world and mathematical problems involving the volume of cylinders and cones.

Solve real-world and mathematical problems involving the volume of spheres.

Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.

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Geometry (all content)

Course: geometry (all content)   >   unit 9.

• Unit test Pythagorean theorem

RWM103: Geometry

Unit 7: right triangle trigonometry.

In this unit we, will explore basic trigonometry. We use trigonometry for several types of measuring techniques, such as calculating the height of a building when you know how far away you are standing from a building and the angle of your gaze to the top. Sailors used trigonometry to determine distances and set their course by using the stars as their guide.

Completing this unit should take you approximately 3 hours.

Upon successful completion of this unit, you will be able to:

• explain and apply the Pythagorean theorem;
• calculate the three basic trigonometric functions of a given angle; and
• calculate the sides of a 45-45-90 or 30-60-90 triangle.

7.1: The Pythagorean Theorem

The Pythagorean theorem allows us to calculate the lengths of the sides of a triangle. We use it frequently to solve for the unknown length of a side of a triangle.

Watch this introductory video.

Watch this video on how to use the Pythagorean theorem to solve for the length of an unknown side of a triangle.

Then, complete these practice problems and check your answers.

7.2: 45-45-90 Triangles

Now that we know the general form of the Pythagorean theorem and how to use it, we can begin exploring special types of triangles and their properties. The first type of triangle we will study is the 45-45-90 right triangle, which is also known as an isosceles triangle.

Read this article and watch the videos to learn about the special properties of isosceles triangles. Pay attention to the examples of finding the length of missing side and solving for unknowns. The article also shows how we can divide squares into two isosceles triangles and then use our knowledge of these triangles to solve for an unknown.

Then, complete review questions 9–12 and check your answers.

7.3: 30-60-90 Right Triangles

The other type of right triangle are 30-60-90 right triangles. These triangles have the special property that their side lengths are always related by a specific ratio.

Read this article and watch the videos. Pay attention to the 30-60-90 theorem, which gives the ratio needed to solve for an unknown side of this type of triangle.

Then, complete review questions 5–8 and check your answers.

7.4: Sine, Cosine, and Tangent

When performing calculations with triangles, as we have seen, we often are interested in ratios between lengths of the sides of the triangle. The study of trigonometry is the study of these ratios.

Read this page and watch the video. This gives a good overview of the three main trigonometric functions: sine, cosine, and tangent. It may be helpful to write the definitions down for yourself to keep track of them.

Then, take this assessment and check your answers.

7.5: Trigonometric Ratios with a Calculator

Finally, we need to learn to use a calculator to solve for sine, cosine, and tangent.

Read this article and watch the videos to learn how to input trigonometric functions into your calculator.

Then, complete review questions 1–3 and check your work to make sure you are able to use your calculator correctly.

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Maneuvering The Middle Llc 2016 Answer Sheet

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Some of the worksheets for this concept are Geometry, Pythagorean theorem work answer key, Linear equations work, Seeing inequalities answer key, Table of contents chapter 2 exploring linear relations 4, Work section 3 2 angles and parallel lines, Lesson plan simple and compound interest, Section volume of cylinders cones and spheres.

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1. Geometry

2. pythagorean theorem worksheet answer key, 3. linear equations worksheet, 4. seeing inequalities answer key, 5. table of contents chapter 2: exploring linear relations (4 ..., 6. worksheet section 3-2 angles and parallel lines, 7. lesson plan -- simple and compound interest, 8. section 8.3: volume of cylinders, cones, and spheres.