Skill: Solve geometric problems using analytic geometry

Use coordinates to turn geometric facts into calculations. For a triangle, you can:

• use slope to test whether sides are parallel or perpendicular;
• use the distance formula to compare side lengths.

Example

Classify the triangle with vertices

$$A(1,2), \qquad B(5,2), \qquad C(5,5).$$

Step 1: Find the slopes

The slope between $(x_1,y_1)$ and $(x_2,y_2)$ is

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

For $\overline{AB}$:

$$m_{AB}=\frac{2-2}{5-1}=\frac{0}{4}=0.$$

So $\overline{AB}$ is horizontal.

For $\overline{BC}$:

$$m_{BC}=\frac{5-2}{5-5}=\frac{3}{0}.$$

The slope is undefined, so $\overline{BC}$ is vertical.

A horizontal line and a vertical line are perpendicular. Therefore,

$$\angle B=90^\circ.$$

The triangle is a right triangle.

Step 2: Find the side lengths

The distance formula is

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$$

For $\overline{AB}$:

$$AB=\sqrt{(5-1)^2+(2-2)^2} =\sqrt{16}=4.$$

For $\overline{BC}$:

$$BC=\sqrt{(5-5)^2+(5-2)^2} =\sqrt{9}=3.$$

For $\overline{AC}$:

$$AC=\sqrt{(5-1)^2+(5-2)^2} =\sqrt{16+9} =\sqrt{25}=5.$$

All three side lengths are different, so the triangle is scalene.

Therefore, the triangle is a right scalene triangle, with the right angle at $B$.