Triangle Area: No Height?--Use the Sine

Learning Resource Type

Lesson Plan

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This lesson will lead students through a review of the proof of the Law of Sines. This proof will remind them that they can use the right triangle relationship for Sine to find the height of a triangle. They will then apply this knowledge to find the area of a triangle when given two sides and an included angle. Finally, they will be asked to find the area when no values are given. This result should produce the Area Formula for a triangle given two sides and the included angle.

This lesson results from the ALEX Resource Gap Project.

Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.37
Derive and apply the formula $A = \frac{1}{2} \cdot ab \cdot \sin(C)$ for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.

Vocabulary

Auxiliary line
Vertex
Perpendicular

Knowledge

Students know:

The auxiliary line drawn from the vertex perpendicular to the opposite side forms an altitude of the triangle.
The formula for the area of a triangle (A = 1/2 bh).
Properties of the sine ratio.

Skills

Students are able to:

Properly label a triangle according to convention.
Perform algebraic manipulations.

Understanding

Students understand that:

Given the lengths of the sides and included angle of any triangle the area can be determined.
There is more than one formula to find the area of a triangle.

Primary Learning Objectives

Students will prove the Law of Sines.

Students will find the area of a triangle given different pieces of information (base and height or two sides and the included angle).

Procedures/Activities

Before:

Time Allotment: 5 minutes

Students will use Law of Sines Proof - Geogebra worksheet to review the proof of the Law of Sines. The teacher will circulate around the room as students work through the proof.

Note: This activity can be done as a whole class activity with the worksheet being projected on the board.

During:

1. Give student pairs a copy of the Area of a Triangle activity. Ask them to complete problems 1-2.

2. Problems 1-2 require students to find the area of triangles given the values of the base and the height. Students will use the formula A=1/2 base x height. Check student work and clear up misconceptions.

3. Next, ask students to complete problems 3-4. These problems do not have a value for the height. The measure of angle A is provided and students will have to use the relationship Sine A = Opposite side (height)/ Hypotenuse (side C). Let students try to solve this problem on their own. If a pair of students do not recognize that they need to use a right triangle relationship lead them to this discovery with questioning.

4. Finally, ask students to complete problem 5. In this problem there are no numerical values given. This problem should lead them to discover that there is a formula for finding the area when given 2 sides and the included angle. Again, circulate among the groups to provide guidance when needed.

5. Allow students to share the formulas they found with the class. Be sure that the entire class can see where the formula for the area comes from. Also be sure that the class is aware that there is more than one way to write the formula based on what is given. The formulas are Area= (1/2)b*c*SinA=(1/2)a*c*SinB=(1/2)a*b*SinC

6. Assign practice problems and allow students to find the area of a triangle using the formula.

After:

Check the answers to the practice problems. Ask students to write a note to a student who might have missed the lesson today. They should explain in detail and in their own words how to find the area of a triangle when two sides and the included angle are given. Tell them that writing the formula will not be enough. They need to explain the formula. Collect these and use this as your exit slip.

Assessment Strategies

Formative Assessment:

Students will self-assess their answers on the Review of the Proof of the Law of Sines - Geogebra worksheet.

Teachers will be able to assess student understanding as they complete the Area of Triangles and Area of Triangles Practice worksheets. Teachers should circulate among students and clear up misconceptions and problems.

Summative Assessment:

Teachers will collect student writing/explanation in the note to a student to assess student understanding of the area formulas when given two sides and an included angle.

Acceleration

Advanced students can be given a triangle with the three side measures given and be asked to find the area of the triangle. They will have to use the Law of Cosines to find an angle and then can use the formula for the area given two sides and the included angle. You could then introduce them to Heron's Formula for finding the area of a triangle given three sides.

Intervention

Students who struggle may need to review the right triangle relationships (SOH-CAH-TOA). They may need to practice finding missing pieces of triangles before attempting to find the area.

Total Duration

31 to 60 Minutes

Background/Preparation

Teachers:

Be familiar with the Geogebra worksheet - students will fill in answers and can check their work. Students do not have to sign in to complete worksheet

Make copies of Area of Triangles activity and Area of Triangles Practice handouts.

Students:

Students should be able to find the missing parts of a triangle using right triangle trigonometry relationships (SOH-CAH-TOA).

Students have been introduced to the Law of Sines.

Students can find the area of a triangle given the base and height of a triangle.

Materials and Resources

Teachers:

Projector

Computer connected to the internet

Geogebra worksheet on proving the Law of Sines (online worksheet can be used as a whole class activity or as individual or student pair activity). Students do not have to sign into Geogebra to complete the worksheet.

Area of Triangles worksheet - One copy per student

Area of Triangles Practice problems - Project on board or make one copy per student

Students:

Computer - one per student or one per pair of students (only if Geogebra activity is individual or pair activity)

Area of Triangles worksheet

Area of Triangles Practice problems

Notebook paper and pencil

Technology Resources Needed

Student computers--one per student or pair (can use one class computer and complete activity as a class)

Geogebra Worksheet on Proof of Law of Sines