MA.912.T.1.4

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.GR.3.4 
• MA.912.GR.4.3
• MA.912.GR.4.4
• MA.912.GR.4.6

Terms from the K-12 Glossary

• Angle
• Area 
• Hypotenuse 
• Right Triangle

Vertical Alignment

Previous Benchmarks

• MA.6.GR.2.1

Next Benchmarks

• MA.912.T.1.5
• MA.912.T.1.6
• MA.912.T.1.7
• MA.912.T.1.8

Purpose and Instructional Strategies

• Within the Geometry course, the expectation is to use angle measures given in degrees and not in radians. Additionally, it is not the expectation for students to master the trigonometric ratios of secant, cosecant and cotangent within this course. 
• It is customary to use Greek letters to represent angle measures (e.g., θ, α, β, γ). 
• Instruction includes making the connection between the formula for the area of a triangle (A = $\frac{\text{1}}{\text{2}}$$b$$h$) and the trigonometric functions (MTR.5.1). 
  • For example, if an acute triangle is given with two sides lengths, $a$ and $b$, and the included angle measure, θ, students can use trigonometric functions to determine the height of the triangle. Students can select side $b$ to be the base and then construct the height of the triangle, $h$, to determine the area. Students should realize that when constructing the height, it creates a right triangle and the length of $h$ can be found by using the sine of the angle θ: $h$ = $a$ sin θ. Students can now use the area formula A = $b$$h$ and substitution to obtain the new formula A = $b$($a$ sin θ). Similarly, students can use the side $a$ as the base and obtain the formula A = $\frac{\text{1}}{\text{2}}$$a$($b$ sin θ).
    
    
    
  • For example, if an obtuse triangle is given with side lengths, $a$ and $b$, and the included obtuse angle measure, θ, students can use trigonometric functions to determine the height of the triangle. Students can select side $b$ to be the base and then construct the height of the triangle, $h$, to determine the area. Students should realize that when constructing the height, it creates a right triangle outside of the given obtuse triangle and the length of $h$ can be found by using the sine of the angle 180 − θ: $h$ = $a$ sin(180 − θ). Students can explore using technology or the unit circle to show that the sin(180 − θ) is equivalent to sin(θ), therefore $h$ = $a$ sin(θ). Students can now use the area formula A = $\frac{\text{1}}{\text{2}}$$b$$h$ and substitution to obtain the new formula A = $\frac{\text{1}}{\text{2}}$$b$($a$ sin θ). Similarly, students can use the side $a$ as the base and obtain the formula A = $\frac{\text{1}}{\text{2}}$$a$($b$ sin θ).
• Instruction includes making the connection to the Law of Sines. (MTR.5.1)
  
  
  
  • For example, if given a triangle with side lengths a, b and c, and opposite angles α, β and γ, then there are three formulas students could use to determine the area: A = $\frac{\text{1}}{\text{2}}$$c$$b$ sin α, A = $\frac{\text{1}}{\text{2}}$$a$$c$ sin β and A = $\frac{\text{1}}{\text{2}}$$b$$a$ sin γ. Students should realize that each of these formulas are equivalent to one another, so that $\frac{\text{1}}{\text{2}}$$c$$b$ sin α = $\frac{\text{1}}{\text{2}}$$a$$c$ sin β = $\frac{\text{1}}{\text{2}}$$b$$a$ sin γ. To make the connection to the Law of Sines, each expression can be divided by $\frac{\text{1}}{\text{2}}$$a$$b$$c$ (Division Property of Equality), to obtain $\frac{\text{sin α}}{\text{<math><mi>a</mi></math>}}$ = $\frac{\text{sin β}}{\text{<math><mi>b</mi></math>}}$ = $\frac{\text{sin γ}}{\text{<math><mi>c</mi></math>}}$.

Common Misconceptions or Errors

• Students may misidentify a side as the height of the triangle and attempt to use A = $\frac{\text{<i>bh</i>}}{\text{2}}$ to find the area. 
• Students may have difficulty determining the height of an obtuse triangle.

Instructional Tasks

• Triangle ABC is given with AB = 22 units, AC = 36 units and the measure of angle α = 29°.
  
  
  
  • Part A. If side AB is the considered the base of the triangle, determine the height of the triangle.
  • Part B. If side AC is the considered the base of the triangle, determine the height of the triangle.
  • Part C. Determine the area of the triangle. Compare your method with a partner.
  • Part D. Determine the length of side BC using the Law of Cosines.
  • Part E. Can you use the answers from Part C and Part D to determine the height of the triangle if BC is considered as the base

Instructional Items

• Find the area of triangle ABC.
  
  
  

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