Lesson 7 Justifying the Laws Solidify Understanding

Jump Start

1.

How are the three squares in this diagram related?

2.

How are the three squares in this diagram related?

3.

Find the missing sides of this triangle using the Law of Sines:

Learning Focus

Derive the Law of Cosines and the Law of Sines.

What happens to the relationship between the areas of the squares on the three sides of a triangle when the triangle is not a right triangle?

Open Up the Math: Launch, Explore, Discuss

The Pythagorean theorem makes a claim about the relationship between the areas of the three squares drawn on the sides of a right triangle: the sum of the area of the squares on the two legs is equal to the area of the square on the hypotenuse. We generally state this relationship algebraically as $a^{2} + b^{2} = c^{2}$ , where it is understood that $a$ and $b$ represent the length of the two legs of the right triangle, and $c$ represents the length of the hypotenuse.

What about non-right triangles? Is there a relationship between the areas of the squares drawn on the sides of a non-right triangle?

The diagram shows an acute triangle with squares drawn on each of the three sides. The three altitudes of the triangle have been drawn and extended through the squares on the sides of the triangle. The altitudes divide each square into two smaller rectangles.

When we refer to $∠ A$ , $∠ B$ , or $∠ C$ we are referring to the angles of the original triangle, even though the altitudes form additional angles at each vertex. Also, the triangle has been labeled in the standard way, with the side opposite $∠ A$ labeled as side $a$ , etc.

1.

Find an expression for the areas of each of the six small rectangles formed by the altitudes. Write these expressions inside each rectangle on the diagram. (Hint: The area of each rectangle can be expressed as the product of the side length of the square and the length of a segment that is a leg of a right triangle. You can use right triangle trigonometry to express the length of this segment.)

2.

Although none of the six rectangles are congruent, there are three pairs of rectangles where each rectangle in the pair has the same area. Using three different colors—red, blue and green—shade pairs of rectangles that have the same area with the same color.

3.

The area of each square is composed of two smaller, rectangular areas of two different colors. Write three different “equations” to represent the areas of each of the squares. For example, you might write $a^{2} = blue + red$ if those are the colors you chose for the areas of the rectangles formed in the square drawn on side $a$ .

4.

Select one of your equations from problem 3, such as $a^{2} = blue + red$ , and use the other two squares to substitute a different expression in for each color. For example, if in your diagram $blue = b^{2} - green$ and $red = c^{2} - green$ , we can write this equation:

$a^{2} = b^{2} - green + c^{2} - green$ or $a^{2} = b^{2} + c^{2} - 2 \cdot green$ .

Write your selected equation in its modified form here:

5.

Since each color is actually a variable representing an area of a rectangle, replace the remaining color in your last equation with the expression that gives the area of the rectangles of that color.

Write your final equation here:

6.

Repeat steps in problems 4 and 5 for the other two equations you wrote in problem 3. You should end up with three different versions of the Law of Cosines, each relating the area of one of the squares drawn on a side of the triangle to the areas of the squares on the other two sides.

a.

$a^{2} =$

b.

$b^{2} =$

c.

$c^{2} =$

7.

What happens to this diagram if angle $C$ is a right angle? (Hint: Think about the altitudes in a right triangle.)

8.

Why do we have to subtract some area from $a^{2} + b^{2}$ to get $c^{2}$ when angle $C$ is less than right?

The Law of Cosines can also be derived for an obtuse triangle by using the altitude of the triangle drawn from the vertex of the obtuse angle, as in the following diagram, where we assume that angle $A$ is obtuse.

9.

Use the diagram to derive one of the forms of the Law of Cosines you wrote above. (Hint: As in the previous task, More Than Right, the length of the altitude can be represented in two different ways, both using the Pythagorean theorem and the portions of side $a$ that form the legs of two different right triangles.)

Ready for More?

Let $c$ be the longest side in a triangle.

For what types of triangles do each of these statements apply?

$a^{2} + b^{2} = c^{2}$ $a^{2} + b^{2} < c^{2}$ $a^{2} + b^{2} > c^{2}$ How do these statements relate to the Law of Cosines derived in the task: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$

What issues do these observations raise if $C$ is a right angle or an obtuse angle?

Takeaways

In addition to the Law of Sines, the Law of Cosines can be used to find missing sides and angles of oblique triangles.

The Law of Cosines can be written in three ways:

Vocabulary

law of cosines
Bold terms are new in this lesson.

Lesson Summary

In this lesson and the previous lesson, we examined two important relationships that exist between the sides and angles of triangles, the Law of Cosines and the Law of Sines. Because of these relationships, we can solve for missing sides and angles in any triangle, not just right triangles.

Retrieval

1.

Tennessee is the $16 th$ largest state in the United States with a population of $6,833,174$ (2019). In 2019 the total population in the U.S. was recorded as $331,814,684$ . The state of Tennessee has a land area of ${41,235 mi}^{2}$ , which makes it $36 th$ in size.

a.

What percent of the people in the United States live in Tennessee?

b.

Calculate the square miles per capita for people living in Tennessee.

2.

Find the missing angle. Do NOT use a calculator.

a.

$\sin θ = \frac{\sqrt{3}}{2}$

b.

$\cos θ = \frac{\sqrt{3}}{2}$

c.

$\tan θ = \frac{\sqrt{3}}{1}$

d.

$\sin θ = \frac{1}{2}$

e.

$\cos θ = \frac{1}{2}$