Illuminations: Squares on a Triangle

Squares on a Triangle
The law of cosines is an extension of the Pythagorean theorem, but seeing how –2ab cos C fits into the picture can be difficult for students. In this lesson, students who understand the Pythagorean theorem and right triangle trigonometry will discover the law of cosines by exploring the areas of squares on the sides of a triangle and their associated "defects."

Learning Objectives
 
By the end of this lesson, students will be able to:
  • Determine whether a triangle is acute, obtuse, or right by comparing the squares of the smaller sides (a2 + b2) to the square of the longer side (c2).
  • Determine a formula for the law of cosines to relate the sides of any type of triangle.
  • Explain how the Law of Cosines is an extension of the Pythagorean theorem.
  • Explain how -2ab cosC is geometrically related to the areas of the squares on sides a, b, and c of a triangle.

Materials
 
Squares on a Triangle Activity Sheet
Squares on a Triangle applet
Pythagorean Review Activity Sheet (optional)
Pythagorean Review applet
Calculator

Instructional Plan
Before starting this lesson, determine if your students need to work through the Pythagorean Review activity. Some things to consider:
  • Are students comfortable with the Pythagorean theorem?
  • Do students recognize the Pythagorean theorem as a geometric relationship among the areas of squares on the sides of a right triangle?

If your students do not possess sufficient understanding of the Pythagorean theorem, have them work on the Pythagorean Review activity sheet. In tandem with the activity sheet, they can investigate the Pythagorean theorem using the Pythagorean Review applet. If students have Internet access at home, you could assign this as independent work to be done before proceeding with this lesson. Students should complete the Pythagorean Review activity sheet individually.

Pythagorean Review Activity Sheet Pythagorean Review Activity Sheet

To proceed with this lesson, distribute the Squares on a Triangle activity sheet.

Squares on a Triangle Activity Sheet Squares on a Triangle Activity Sheet

This activity sheet must be completed using a computer that has Internet access, since it involves accessing the Squares on a Triangle applet. In a computer lab setting, pair students so that they cay work on the activity sheet together. Have one student read the activity sheet while the other manipulates the applet. Partners may switch roles after Question 5.

Squares on a Triangle Applet Squares on a Triangle Applet

If students are not used to doing this type of discovery learning, encourage their creativity. Tell them that you are looking for their thoughts and intuitions and not so much concerned that they get the right answer the first time through.

 

Selected Answers to the Squares on a Triangle Activity Sheet

1. Answers will vary, but Area Square I + Area Square II should be larger than Area Square III for acute triangle, and smaller for obtuse triangles.

(Students may have difficulty understanding what exactly is meant by "compare," so for students are having difficulty getting started, simply asking, "Is one bigger than the other?" may suffice.)

2. In an acute triangle, a2 + b2 will be greater than c2.

In an obtuse triangle, a2 + b2 will be less than c2.

6. For acute triangles, students should realize that ASquare I + ASquare II – ADefect I – ADefect II = ASquare III.

For obtuse triangles, students should realize that ASquare I + ASquare II + ADefect I + ADefect II = ASquare III.

(If students are struggling at this point, remind them that they are attempting to find some combination of the areas of Square I, Square II, Defect I, and Defect II that will equal the area of Square III.

For the acute triangle, as a further hint, point out that the defects overlap the squares on each side. Also, since in acute triangles, the sum of the square areas on the smaller two sides is larger than the area of the square on the largest side, the defects need to be subtracted from the total combination in order to make it equal to the area of Square III.

For the obtuse triangle, notice that the defects do not overlap the squares on each side this time; hence, the defect areas needed to be added to the areas of Squares I and II.)

Depending upon the progess of the students, Questions 7 through 9 might be more appropriate for a guided class discussion, leaving Questions 10 through 13 as individual or partner work.

7. Since the area of a rectangle is length times width, the area is a · x.

8. x = b cos C

(Students struggling at this point should have the right triangle containing sides x and b, as well as ∠C.)

9. and 10. Area = a · x = ab cos C.

12. The area of Square III equals the areas of Square I and Square II minus the areas of Defect I and Defect II.

13. c2 = a2 + b2 – 2ab cos C.

Questions for Students
 
What we mean when we say that the law of cosines extends the Pythagorean theorem to any type of triangle?
[Whereas the Pythagorean theorem relates the sides of a right triangle using the areas of squares on the sides of the triangle, the law of cosines works for any type of triangle by adjusting the areas of the squares on the sides.]

Why do you think that the area of the rectangles subtracted in the law of cosines are called "defects"?

[These areas represent the amount of area that the squares on the two shorter sides of a triangle overestimate or underestimate the area of the square on the longest side. We must account for these areas by either adding or subtracting them.]

The Law of Cosines states that c2 = a2 + b2 – 2ab cos C, where a, b, and c are sides of a triangle. From your previous work, where does angle C need to be in the triangle?

[Angle C must be opposite c, the longest side of the triangle. It must be opposite the side whose square has an area equal to the sum of the other two squares and defects.]

Could you use the Law of Cosines to determine the angles of a triangle if you knew all three side lengths?

[Yes, by solving for the cosine of angle C and then applying the inverse cosine function. It is usually preferable to encourage students to solve the law of cosines for this quantity algebraically, rather than having them memorize this form.]

Assessment Options
 
  1. The following problems can be used to assess students’ ability to use the law of cosines.
    • A triangle has sides of length 6 cm and 8 cm. The angle between the two known sides is 75°. Find the length of the unknown side. [c ≈ 8.669104433…]
    • A triangle has sides of length 14, 19, and 27 units. What is the area of each defect on the two smaller sides? [The area of the square on the largest side is 272 = 729 square units. Subtracting the other two areas (142 + 192)yields 172 square units, meaning that the area of each defect is 86 square units.]
    • A triangle has sides of length 9 feet, 13 feet, and 17 feet. How big is the angle opposite the side that is 17 feet long? How big is the angle opposite the side that is 9 feet long? Find the size of the angle opposite the side of length 13 feet without using the law of cosines. [C ≈ 99.594° and A ≈ 31.467°. The measure of B=180 – 99.594 – 31.467 = 48.939°.]
    • Determine AC if AB = 10, BC = 7, and m∠A = 20°. [The law of cosines can be used, once it is recognized that ∠A is opposite side BC. When substituted, a quadratic equation results: 72 = x2+102 – 2x(10 cos 20), which yields x = 15.5044 or x = 3.2894. If your students have already studied the ambiguous triangle case using the law of sines, the law of cosines is a good alternative for quickly seeing both cases.]
  2. To assess the level of student comprehension, ask students to create two different questions involving the ideas studied. Request one basic question and one advanced question. Ask students to submit their problems before leaving class as an "exit card." Review the level of questions to determine the level of student understanding.

Extensions
 
  1. Ask students to consider the implications when the sum of the squares on the smaller sides of a triangle is larger than the square on the longest side.
  2. In this lesson, students learned that the defect areas must be subtracted if the triangle is acute but added if the triangle is obtuse. Ask students to explain why the same law of cosines formula works for both acute and obtuse triangles, but you don't change subtraction to addition in the formula!

Teacher Reflection
 
  • Was students’ level of enthusiasm and involvement high or low? Explain why.
  • Was your lesson developmentally appropriate? If not, what was inappropriate? What can you do to change it?
  • What problems did students encounter when calculating the areas of the defects? How could you change the presentation of this lesson to increase student understanding regarding the area of the defects?
  • By the end of the lesson, did students understand that the law of cosines could be used to find the length of any side of a triangle given the two sides and the included angle?
  • By the end of the lesson, did students understand that the law of cosines extends the Pythagorean theorem to any type of triangle?

NCTM Standards and Expectations
 
Algebra 9-12
  1. Use symbolic algebra to represent and explain mathematical relationships.
Geometry 9-12
  1. Use geometric models to gain insights into, and answer questions in, other areas of mathematics.
  2. Use trigonometric relationships to determine lengths and angle measures.
This lesson prepared by Curtis James.
  
2 periods   

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Principles and Standards for School Mathematics

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