# Right Triangles and Trigonometry

## Common Core Standards

### Core Standards

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• A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

• N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

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• 8.EE.A.2

## Criteria for Success

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2. Treat the coefficients of radicals like the coefficient of like terms when you add or subtract.
3. Identify when you can create like terms with radicands to make it possible to add or subtract. For example: ${\sqrt{8}+\sqrt{2}=\sqrt{4\cdot2}+\sqrt{2}=2\sqrt{2}+\sqrt{2}=3\sqrt{2}}$
4. Calculate the perimeter of right triangles with radical side lengths.

## Tips for Teachers

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• This lesson extends work done in Algebra 1. The properties of radicals should be familiar to students but will need some review.
• The focus of this lesson is on working with numeric radical expressions, but students should practice with algebraic radical expressions as well.

## Anchor Problems

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### Problem 1

Simplify the following:

${2\sqrt{3}+3\sqrt{3}}$

${3\sqrt{12}-\sqrt{27}}$

### Problem 2

Is this statement always, sometimes, or never true? Explain your reasoning.

${\sqrt{a}+\sqrt b=\sqrt{a+b}}$

### Problem 3

Find the perimeter of a right triangle with legs of length of ${6\sqrt 2}$ and ${\sqrt 3}$.

## 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 where students need to find the value of a leg or hypotenuse and then calculate the perimeter of the triangle, similar to Anchor Problem #3.
• Include problems with special right triangles.
• Include problems where students need to compare the properties of multiplication of radicals with the properties of addition/subtraction of radicals.

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### Problem 1

Simplify ${\sqrt{8}+5\sqrt{2}}$.

#### References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 23Exit Ticket, Question #2

Geometry > Module 2 > Topic D > Lesson 23 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

Write a radical addition or subtraction problem that cannot be simplified, and explain why it cannot be simplified.

#### References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 23Exit Ticket, Question #3

Geometry > Module 2 > Topic D > Lesson 23 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..