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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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  1. Use the area formula for triangles to determine missing information in non-right triangles.
  2. Identify that an auxiliary altitude is necessary in non-right triangles to represent the height.
  3. Write the measurement of the height of a non-right triangle in terms of sine of the angle opposite to the height. 
  4. Assign the variables $$A$$, $$B$$, and $$C$$ to each of the vertices of a non-right triangle. 
  5. Assign the variables $$a$$$$b$$, and $$c$$ to each side opposite vertices $$A$$, $$B$$, and $$C$$ respectively in non-right triangles.
  6. Generalize the area formula for non-right triangles as $$\mathrm{Area}=\frac{1}{2}ab\mathrm{sin}(C)$$.
  7. 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

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Problem 2

Write a formula for the area of this non-right triangle.

Guiding Questions

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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.

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