# MA.912.T.1.4

Solve mathematical problems involving finding the area of a triangle given two sides and the included angle.

### Clarifications

Clarification 1: Problems include right triangles, heights inside of a triangle and heights outside of a triangle.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Trigonometry
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Angle
• Area
• Hypotenuse
• Right Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students learned how to find the area of a triangle using a formula and its connection to the area of a rectangle. In Geometry, students find the area of a triangle when given the length of two sides and the included angle. In later courses, students will use further trigonometric relationships and identities to determine information that is needed to find areas of triangles.
• Within the Geometry course, the expectation is to use angle measures given in degrees and not in radians. Additionally, it is not the expectation for students to master the trigonometric ratios of secant, cosecant and cotangent within this course.
• It is customary to use Greek letters to represent angle measures (e.g., θ, α, β, γ).
• Instruction includes making the connection between the formula for the area of a triangle (A = $\frac{\text{1}}{\text{2}}$$b$$h$) and the trigonometric functions (MTR.5.1)
• For example, if an acute triangle is given with two sides lengths, $a$ and $b$, and the included angle measure, θ, students can use trigonometric functions to determine the height of the triangle. Students can select side $b$ to be the base and then construct the height of the triangle, $h$, to determine the area. Students should realize that when constructing the height, it creates a right triangle and the length of $h$ can be found by using the sine of the angle θ: $h$ = $a$ sin θ. Students can now use the area formula A = $b$$h$ and substitution to obtain the new formula A = $b$($a$ sin θ). Similarly, students can use the side $a$ as the base and obtain the formula A = $\frac{\text{1}}{\text{2}}$$a$($b$ sin θ).

• For example, if an obtuse triangle is given with side lengths, $a$ and $b$, and the included obtuse angle measure, θ, students can use trigonometric functions to determine the height of the triangle. Students can select side $b$ to be the base and then construct the height of the triangle, $h$, to determine the area. Students should realize that when constructing the height, it creates a right triangle outside of the given obtuse triangle and the length of $h$ can be found by using the sine of the angle 180 − θ: $h$ = $a$ sin(180 − θ). Students can explore using technology or the unit circle to show that the sin(180 − θ) is equivalent to sin(θ), therefore $h$ = $a$ sin(θ). Students can now use the area formula A = $\frac{\text{1}}{\text{2}}$$b$$h$ and substitution to obtain the new formula A = $\frac{\text{1}}{\text{2}}$$b$($a$ sin θ). Similarly, students can use the side $a$ as the base and obtain the formula A = $\frac{\text{1}}{\text{2}}$$a$($b$ sin θ).
• Instruction includes making the connection to the Law of Sines. (MTR.5.1)

• For example, if given a triangle with side lengths a, b and c, and opposite angles α, β and γ, then there are three formulas students could use to determine the area: A = $\frac{\text{1}}{\text{2}}$$c$$b$ sin α, A = $\frac{\text{1}}{\text{2}}$$a$$c$ sin β and A = $\frac{\text{1}}{\text{2}}$$b$$a$ sin γ. Students should realize that each of these formulas are equivalent to one another, so that $\frac{\text{1}}{\text{2}}$$c$$b$ sin α = $\frac{\text{1}}{\text{2}}$$a$$c$ sin β = $\frac{\text{1}}{\text{2}}$$b$$a$ sin γ. To make the connection to the Law of Sines, each expression can be divided by $\frac{\text{1}}{\text{2}}$$a$$b$$c$ (Division Property of Equality), to obtain $\frac{\text{sin α}}{\text{a}}$ = $\frac{\text{sin β}}{\text{b}}$ = $\frac{\text{sin γ}}{\text{c}}$.

### Common Misconceptions or Errors

• Students may misidentify a side as the height of the triangle and attempt to use A = $\frac{\text{bh}}{\text{2}}$ to find the area.
• Students may have difficulty determining the height of an obtuse triangle.

• Triangle ABC is given with AB = 22 units, AC = 36 units and the measure of angle α = 29°.

• Part A. If side AB is the considered the base of the triangle, determine the height of the triangle.
• Part B. If side AC is the considered the base of the triangle, determine the height of the triangle.
• Part C. Determine the area of the triangle. Compare your method with a partner.
• Part D. Determine the length of side BC using the Law of Cosines.
• Part E. Can you use the answers from Part C and Part D to determine the height of the triangle if BC is considered as the base

### Instructional Items

Instructional Item 1
• Find the area of triangle ABC.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1202340: Precalculus Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plan

The Seven Circles Water Fountain:

Students will apply concepts related to circles, angles, area, and circumference to a design situation.

Type: Lesson Plan

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)