Unlock Areas with Trig: Mastering the Area Rule

Algebra 2 Grades High School 12:13 Video
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Lesson Description

Learn how to calculate the area of a triangle when you know two sides and the included angle using the area rule. This lesson covers the formula, examples, and problem-solving techniques.

Video Resource

Area Rule Algebra 2

Kevinmathscience

Duration: 12:13
Watch on YouTube

Key Concepts

  • Area Rule Formula (Area = 1/2 * a * b * sin(C))
  • Identifying the included angle and adjacent sides
  • Using the Sine Rule to find missing sides or angles (if needed)
  • Applying the sum of angles in a triangle rule

Learning Objectives

  • Students will be able to apply the area rule formula to calculate the area of a triangle given two sides and the included angle.
  • Students will be able to identify the appropriate sides and angle for use in the area rule formula.
  • Students will be able to use the sine rule in conjunction with the area rule when necessary to solve for missing information.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the traditional area formula (1/2 * base * height) and discuss its limitations when the height is unknown. Introduce the area rule as an alternative method.
  • Video Presentation (15 mins)
    Play the Kevinmathscience video 'Area Rule Algebra 2'. Instruct students to take notes on the formula and the examples provided.
  • Guided Practice (15 mins)
    Work through example problems similar to those in the video, emphasizing the correct identification of sides 'a' and 'b', and angle 'C'. Guide students through the calculation steps.
  • Independent Practice (10 mins)
    Assign practice problems for students to solve individually. Circulate to provide assistance as needed.
  • Advanced Problem Solving (5 mins)
    Introduce scenarios where the sine rule needs to be implemented before area rule can be used.

Interactive Exercises

  • Triangle Area Calculator
    Provide students with a worksheet containing various triangle dimensions (two sides and the included angle). Have them calculate the areas using the area rule. Increase difficulty by including triangles where an intermediate step using the sine rule is needed before calculating the area.
  • Error Analysis
    Present students with solved problems containing common errors (e.g., incorrect angle selection, incorrect formula application). Ask them to identify and correct the errors.

Discussion Questions

  • When is it most useful to use the area rule instead of the traditional area formula?
  • How does changing the size of the angle C affect the area of the triangle, assuming the sides a and b remain constant?
  • Explain the difference between using the sine rule and area rule

Skills Developed

  • Problem-solving
  • Critical Thinking
  • Application of Trigonometric Formulas
  • Using formulas to solve for unknowns

Multiple Choice Questions

Question 1:

The area rule formula for a triangle is:

Correct Answer: Area = 1/2 * a * b * sin(C)

Question 2:

In the area rule, 'C' represents:

Correct Answer: The included angle between sides 'a' and 'b'

Question 3:

When should you use the area rule instead of the standard area formula (1/2 * base * height)?

Correct Answer: When you know two sides and the included angle

Question 4:

If a = 5, b = 8, and C = 30 degrees, what is the area of the triangle?

Correct Answer: 10

Question 5:

You are given a triangle with angle C = 60 degrees, side a = 7, but side b is unknown. However you are given side c = 9 and angle A = 40 degrees. What is the first step to find the area?

Correct Answer: Calculate side b using the Law of Sines

Question 6:

What is the area of a triangle where side a = 10, side b = 12, and angle C = 90 degrees?

Correct Answer: 60

Question 7:

If the area of a triangle is 24, side a is 6, and angle C is 30 degrees, what is the length of side b?

Correct Answer: 16

Question 8:

Which of the following is NOT needed when using the area rule?

Correct Answer: Height of the triangle

Question 9:

Two sides of a triangle are 15 and 20, and the included angle is 45 degrees. What is the area of the triangle?

Correct Answer: 150

Question 10:

If side a = 4, side b = 7, and angle C = 75 degrees, find the area of the triangle.

Correct Answer: Approximately 27.04

Fill in the Blank Questions

Question 1:

The area rule formula is Area = 1/2 * a * b * __________(C).

Correct Answer: sin

Question 2:

In the area rule, the angle C must be the __________ angle between sides a and b.

Correct Answer: included

Question 3:

If you know all three angles of a triangle and one side, use the __________ rule to find the other sides.

Correct Answer: sine

Question 4:

To calculate the area, you must use the angle that is __________ by the two known sides.

Correct Answer: formed

Question 5:

The angles in a triangle must add up to __________ degrees

Correct Answer: 180

Question 6:

If a = 9, b = 11, and C = 45 degrees, the area of the triangle is approximately __________.

Correct Answer: 34.65

Question 7:

If Area = 30, a = 10, and b = 8, sin(C) = __________. (Express as a decimal).

Correct Answer: 0.75

Question 8:

The area rule is useful when you do not know the __________ of a triangle.

Correct Answer: height

Question 9:

When you need to use sine rule before you can use area rule, you can say area rule implementation is a __________ implementation

Correct Answer: secondary

Question 10:

When reporting area, you must include __________ units in the answer

Correct Answer: squared

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

Teaching Materials

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