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
If your students do not possess sufficient understanding of the Pythagorean theorem, have them work on the Pythagorean Review Activity Sheet.
Pythagorean Review Activity Sheet
In tandem with the activity sheet, they can investigate the Pythagorean theorem using the Pythagorean Review Applet.
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
To proceed with this lesson, distribute the 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
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 progress 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
1. 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.]
2. 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.]
3. 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.]
4. 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.]
- 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
- 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?
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.
NCTM Standards and Expectations
- Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
- Use trigonometric relationships to determine lengths and angle measures.
- Use geometric models to gain insights into, and answer questions in, other areas of mathematics.