Right Triangles and Trigonometry
Lesson 16
Objective
Derive the area formula for any triangle in terms of sine.
Common Core Standards
Core Standards
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G.SRT.D.9 — Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
Foundational Standards
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6.G.A.1
Criteria for Success
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- Use the area formula for triangles to determine missing information in non-right triangles.
- Identify that an auxiliary altitude is necessary in non-right triangles to represent the height.
- Write the measurement of the height of a non-right triangle in terms of sine of the angle opposite to the height.
- Assign the variables $$A$$, $$B$$, and $$C$$ to each of the vertices of a non-right triangle.
- Assign the variables $$a$$, $$b$$, and $$c$$ to each side opposite vertices $$A$$, $$B$$, and $$C$$ respectively in non-right triangles.
- Generalize the area formula for non-right triangles as $$\mathrm{Area}=\frac{1}{2}ab\mathrm{sin}(C)$$.
- Verify that the formula works for any non-right triangle once $$a$$, $$b$$, and $$c$$ sides and $$A$$, $$B$$, and $$C$$ angles are defined.
Tips for Teachers
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This post on Math Central is helpful in developing knowledge on the area formula in terms of sine for triangles.
Anchor Problems
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Problem 1
Find the area of the following triangle.
Guiding Questions
Problem 2
Write a formula for the area of this non-right triangle.
Guiding Questions
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 the students are given the area and need to find missing information about the non-right triangle.
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EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic B > Lesson 7
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EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 31
Target Task
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Explain why $${\frac{1}{2}ab\mathrm{sin}(C)}$$ gives the area of a triangle with sides $$a$$ and $$b$$ and included angle $$C$$.
References
Precalculus and Advanced Topics > Module 4 > Topic B > Lesson 7 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..
Modified by Fishtank Learning, Inc.