Lesson 8 Triangle Area by Trigonometry Practice Understanding

Learning Focus

Apply the Law of Cosines and the Law of Sines to solve problems.

How can I find the area of a triangle when I don’t know both of the lengths of the base and height?

Open Up the Math: Launch, Explore, Discuss

Find the area of the following two triangles. To find needed information use any of the strategies or procedures you have developed:

draw an altitude as an auxiliary line
use right triangle trigonometry
use the Pythagorean theorem
use the Law of Sines or the Law of Cosines to find needed information

1.

Find the area of this triangle.

2.

Find the area of this triangle.

Jumal and Jabari are helping Jumal’s father with a construction project. He needs to build a triangular frame as a piece to be used in the whole project, but he has not been given all the information he needs to cut and assemble the sides of the frame. He is even having a hard time envisioning the shape of the triangle from the information he has been given.

Here is the information about the triangle that Jumal’s father has been given.

Side $a = 10.00 meters$
Side $b = 15.00 meters$
Angle $A = 40.0 °$

Jumal’s father has asked Jumal and Jabari to help him find the measure of the other two angles and the missing side of this triangle.

3.

Carry out each student’s strategy as described below. Then draw a diagram showing the shape and dimensions of the triangle that Jumal’s father should construct. (Note: Provide measurements at an appropriate level of accuracy for the given information.)

a.

Jumal’s Approach

Find the measure of angle $B$ using the Law of Sines

b.

Find the measure of the third angle $C$

c.

Find the measure of side $c$ using the Law of Sines

d.

Draw the triangle

e.

Jabari’s Approach

Solve for $c$ using the Law of Cosines

f.

Jabari is surprised that this approach leads to a quadratic equation, which he solves with the quadratic formula. He is even more surprised when he finds two reasonable solutions for the length of side $c$ .

Draw both possible triangles and find the two missing angles of each using the Law of Sines

4.

A city plans to purchase a triangular plot of land from a farmer to create a park. They need to determine the area of the land in order to offer a fair purchase price. (Land is typically sold by the acre and $1 acre = 43, 560 {ft}^{2}$ .) Surveyors have provided the measurements given in the diagram. Use this information to calculate the area of the plot of land in acres.

Ready for More?

The following diagram shows two different triangles with the same side-side-angle measures. Find the area of both triangles.

Takeaways

The area of a triangle can be found by using the formula if , and by using the formula if .

The Ambiguous Case of the Law of Sines occurs when because .

To avoid missing a possible solution for an oblique triangle under these conditions, I can .

Vocabulary

Ambiguous Case of the Law of Sines
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we practice using the Law of Sines and the Law of Cosines and applied them to a practical situation. We learned that when we are given SSA information about an oblique triangle there are two possible triangles that satisfy these conditions. The Law of Sines doesn’t give us both solutions, but the Law of Cosines will. We also developed a new formula for finding the area of a triangle when we are given SAS information about the triangle.

Retrieval

1.

Alaska has an area of $665,384.14 square miles$ . However, $14.24 %$ is covered in water. What is the land area of Alaska?

2.

Indicate whether you would use the Law of Sines or the Law of Cosines to solve the triangles.