Your friend is creating a new board game that involves several different triangle shaped pieces. However, the game requires accurate measurements of several different pieces that all have to fit together. She brings some of the pieces to you and asks if you can verify that her measurements of the pieces' side lengths and angles are correct.

You take out the first piece. According to your friend, the piece has sides of length 4 in, 5 in and 7 in, and the angle between the side of the length 4 and the side of length 5 is

### Analyzing Triangles

Our extension of the analysis of triangles draws us naturally to oblique triangles.

The Law of Cosines can be used to verify that drawings of oblique triangles are accurate. In a right triangle, we might use the Pythagorean Theorem to verify that all three sides are the correct length, or we might use trigonometric ratios to verify an angle measurement. However, when dealing with an obtuse or acute triangle, we must rely on the Law of Cosines.

For the following problems, let's use the Law of Cosines

1. In

We will use the Law of Cosines to check whether or not

Since

For some situations, it will be necessary to utilize not only the Law of Cosines, but also the Pythagorean Theorem and trigonometric ratios to verify that a triangle or quadrilateral has been drawn accurately.

2. A builder received plans for the construction of a second-story addition on a house. The diagram shows how the architect wants the roof framed, while the length of the house is 20 ft. The builder decides to add a perpendicular support beam from the peak of the roof to the base. He estimates that new beam should be 8.3 feet high, but he wants to double-check before he begins construction. Is the builder’s estimate of 8.3 feet for the new beam correct? If not, how far off is he?

**I**f we knew either

Now that we know

Yes, the builder’s estimate of 8.3 feet for the support beam is accurate.

#### In △CIR,c=63,i=52 , and r=41.9 . Find the measure of all three angles.

### Examples

#### Example 1

Earlier, you were asked if there was a way to determine if your friend's game piece has the correct measurements.

Since your friend is certain of the lengths of the sides of the triangle, you should use those as the known quantities in the Law of Cosines and solve for the angle:

So as it turns out, your friend is rather close. Her measurements were probably slight inaccurate because of her round off from the protractor.

#### Example 2

Find

First, find

#### Example 3

Find

#### Example 4

Use the Law of Cosines to determine whether or not the following triangle is drawn accurately. If not, determine how far the measurement of side "d" is from the correct value.

To determine this, use the Law of Cosines and solve for

### Review

- If you know the lengths of all three sides of a triangle and the measure of one angle, how can you determine if the triangle is drawn accurately?

Determine whether or not each triangle is labelled correctly.

Determine whether or not each described triangle is possible. Assume angles have been rounded to the nearest degree.

- In
△BCD , b=4, c=4, d=5, andm∠B=51∘ . - In
△ABC , a=7, b=4, c=9, andm∠B=34∘ . - In
△BCD , b=3, c=2, d=7, andm∠D=138∘ . - In
△ABC , a=8, b=6, c=13.97, andm∠C=172∘ . - In
△ABC , a=4, b=4, c=9, and \begin{align*}m\angle B=170^\circ\end{align*}. - In \begin{align*}\triangle BCD\end{align*}, b=3, c=5, d=4, and \begin{align*}m\angle C=90^\circ\end{align*}.
- In \begin{align*}\triangle ABC\end{align*}, a=8, b=3, c=6, and \begin{align*}m\angle A=122^\circ\end{align*}.
- If you use the Law of Cosines to solve for \begin{align*}m\angle C\end{align*} in \begin{align*}\triangle ABC\end{align*} where a=3, b=7, and c=12, you will an error. Explain why.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.3.