Right Triangles and Trigonometry
Lesson 2
Objective
Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.
Common Core Standards
Core Standards
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G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Foundational Standards
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8.EE.A.2
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8.G.B.6
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8.G.B.7
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8.G.B.8
Criteria for Success
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- Define the Pythagorean theorem as, “If a triangle is a right triangle, then the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.”
- Define the converse of the Pythagorean theorem as, “If the sum of the squares of two shorter sides is equal to the square of the third side, then the triangle is a right triangle.”
- Prove the Pythagorean theorem using the similarity of the two triangles formed by subdividing the larger triangle using an altitude.
- Use the Pythagorean theorem to find missing measures in right triangles.
- Use altitudes and the Pythagorean theorem to identify missing measures (directly or algebraically) and in the development of proofs.
Tips for Teachers
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- This lesson formalizes students’ understanding of the Pythagorean theorem from Grade 8 Geometry.
- Anchor Problem #1 will require the teacher to do a significant amount of modeling to prove the Pythagorean theorem and its converse. Reference EngageNY, Geometry, Module 2, Lesson 24: Teacher Version for more guidance on this.
- Students may need to review simplifying radicals before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.
Anchor Problems
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Problem 1
Given the subdivided right triangle below, show that $${a^2+b^2=c^2}$$.
- Write an expression in terms of $$c$$ for $$x$$ and $$y$$.
- Write a similarity statement for the three right triangles in the diagram.
- Write a ratio that shows the relationship between side lengths of two of the triangles.
- Prove the Pythagorean theorem.
- Prove the converse of the Pythagorean theorem.
Guiding Questions
References
Geometry > Module 2 > Topic D > Lesson 24 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Problem 2
Find $${RS}$$.
Guiding Questions
References
8.3 Using Similar Right Triangles is made available by CK-12 under the CC BY-NC 3.0 license. © CK-12 Foundation 2017. Visit them at ck12.org. Accessed Feb. 28, 2018, 11:10 a.m..
Problem Set
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
- Include problems with descriptions and note ratios of right triangles with an altitude drawn.
- Include more basic practice with Pythagorean theorem, such as Illustrative Mathematics, “8.G.B” specifically “Applying the Pythagorean Theorem in a Mathematical Context”.
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Mathematics Vision Project: Secondary Mathematics Two Module 6: Similarity and Right Triangle Trigonometry
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Lesson 7 "Pythagoras by Proportions" (p. 39-43)
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Illustrative Mathematics Finding the Area of an Equilateral Triangle
Target Task
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Problem 1
In each row in the table below are the lengths of the legs and hypotenuse of different right triangles. Find the missing side lengths in each row in simplest radical form.
| $${\mathrm{Leg}_1}$$ | $${\mathrm{Leg}_2}$$ | Hypotenuse |
| 15 | 25 | |
| 15 | 36 | |
| 3 | 7 | |
| 100 | 200 |
References
Geometry > Module 2 > Topic D > Lesson 24 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Problem 2
In right triangle $${ABC}$$ with $${\angle C}$$ a right angle, an altitude of length $$h$$ is dropped to side $${\overline{AB}}$$ into segments of length $$x$$ and $$y$$. Use the Pythagorean theorem to show $$h^2=xy$$.
References
Geometry > Module 2 > Topic D > Lesson 24 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..