Lesson Objectives

- Learn how to find the area of a triangle using determinants

## How to Find the Area of a Triangle Using Determinants

In this lesson, we want to learn how to find the area of a triangle using determinants. A common application of matrices and determinants is being able to find the area of a triangle when the vertices are given to us as points in the coordinate plane. Suppose we have a triangle with the following vertices: $$(x_1, y_1)$$ $$(x_2, y_2)$$ $$(x_3, y_3)$$ The area of a triangle is given using the following formula: $$\text{Area}=\pm \frac{1}{2}\left| \begin{array}{ccc}x_{1}&y_{1}&1\\ x_{2}& y_{2}& 1\\ x_{3}& y_{3}& 1\end{array}\right|$$ The plus or minus sign out in front just tells us to make sure our answer is positive. It wouldn't make any sense to have a negative area. Therefore, the result of this formula should be either 0 or some positive number. Let's look at an example.

Example #1: Find the area of the triangle whose vertices are given below. $$(1, 5), (3, -2), (6, -4)$$ We can label the points in any order, just make sure to be consistent. $$\text{Point 1}:(1,5)$$ $$\text{Point 2}:(3,-2)$$ $$\text{Point 3}:(6,-4)$$ Now let's plug into our formula. $$\text{Area}=\pm \frac{1}{2}\left| \begin{array}{ccc}x_{1}&y_{1}&1\\ x_{2}& y_{2}& 1\\ x_{3}& y_{3}& 1\end{array}\right|$$ $$\text{Area}=\pm \frac{1}{2}\left| \begin{array}{ccc}1&5&1\\ 3& -2& 1\\ 6& -4& 1\end{array}\right|$$ $$\text{Area}=\pm \frac{1}{2}\cdot 17$$ $$\text{Area}=\frac{17}{2}$$ Our answer is 17/2 square units.

Example #1: Find the area of the triangle whose vertices are given below. $$(1, 5), (3, -2), (6, -4)$$ We can label the points in any order, just make sure to be consistent. $$\text{Point 1}:(1,5)$$ $$\text{Point 2}:(3,-2)$$ $$\text{Point 3}:(6,-4)$$ Now let's plug into our formula. $$\text{Area}=\pm \frac{1}{2}\left| \begin{array}{ccc}x_{1}&y_{1}&1\\ x_{2}& y_{2}& 1\\ x_{3}& y_{3}& 1\end{array}\right|$$ $$\text{Area}=\pm \frac{1}{2}\left| \begin{array}{ccc}1&5&1\\ 3& -2& 1\\ 6& -4& 1\end{array}\right|$$ $$\text{Area}=\pm \frac{1}{2}\cdot 17$$ $$\text{Area}=\frac{17}{2}$$ Our answer is 17/2 square units.

#### Skills Check:

Example #1

Find the area. $$(-2, 4), (6, 7), (3, -1)$$

Please choose the best answer.

A

19/2 square units

B

33/2 square units

C

77 square units

D

55/2 square units

E

13 square units

Example #2

Find the area. $$(2, 1), (1, 4), (5, -3)$$

Please choose the best answer.

A

3 square units

B

5/2 square units

C

2 square units

D

7/2 square units

E

19/2 square units

Example #3

Find the area. $$(7, -5), (3, 4), (9, 8)$$

Please choose the best answer.

A

16 square units

B

3/2 square units

C

9/5 square units

D

35 square units

E

17 square units

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