Question

Find the area of the triangle with vertices A(-3, 2), B(1, - 2). and C(1, 3).​

Answer 1

10 units squared

Explanation:

Main steps

Step 1. Identify which leg to use as a base

Step 2. Calculate the base length

Step 3. Calculate the height

Step 4. Calculate the area

Step 1. Identify which leg to use as a base

We are given that the shape is a triangle. The formula for the area of a triangle is
`A_(triangle)=(1)/(2)bh` where b is the "base" of a triangle, and h is the "height".

One common misconception is that the base must be at the bottom (although it is easy to think of it that way). Rotating the shape, any of the three sides could be at the bottom, and thus any of the three sides could be the base.

The "height" of the triangle depends on which leg is being used as a base. The height for a given base is always the shortest distance from the vertex opposite the base to the line that contains the base. This shortest distance always ends up being the line segment that is perpendicular to the base.

In this case, since side BC is along the gridlines, it will be easiest to use side BC as the base, because the perpendicular line for the height will be along gridlines. The height is the distance from A to the side BC (which is a line segment between (-3,2) and (1,2)).

Step 2. Calculate the base length

The base length is the distance between the two endpoints of the line segment. The distance formula is
`D=√((x_2-x_1)^2+(y_2-y_1)^2)`

Using Point B as Point 1, and Point C as Point 2:

`D=√(((1)-(1))^2+((3)-(-2))^2)`

`D=√((0)^2+(5)^2)`

`D=√(0+25)`

`D=√(25)`

`D=5` units

So the base length is 5 units

b=5 units

Since these point are on a graph, it can be verified by counting the squares between the two points, however if work needs to be shown, the Distance formula is the "work".

Step 3. Calculate the height

The "height" for the base from step 2 is a line segment between (-3,2) and (1,2). This distance can again be found using the distance formula:

Using Point A as Point 1, and the point (1,2) on BC as Point 2:

`D=√(((-3)-(1))^2+((2)-(2))^2)`

`D=√((-4)^2+(0)^2)`

`D=√(16+0)`

`D=√(16)`

`D=4` units

So the height for the base from Step 2 is 4 units.

h=4 units

Again, these point are on a graph, so this answer can be verified by counting the squares between the two points.

Step 4. Calculate the area

Using the formula for the area of a triangle discussed in Step 1, the area can be calculated:

`A_(triangle)=(1)/(2)(5~\text{units})(4~\text{units})`

`A_(triangle)=10 \text{ units}^2`