Solve geometric problems using coordinate methods

Detailed Explanation: Solve geometric problems using coordinate methods

Coordinates let us turn a picture into numbers. The main steps are:

1. Plot or inspect the points.
2. Compare their horizontal and vertical changes.
3. Use a formula or relationship to answer the geometry question.
4. Interpret the result in the figure.

Example: Determine whether the triangle with vertices
$A(-1,1)$, $B(3,1)$, and $C(3,4)$ is a right triangle. Then find its area.

Step 1: Look at the points

Points $A$ and $B$ have the same $y$-coordinate:

So $\overline{AB}$ is horizontal.

Points $B$ and $C$ have the same $x$-coordinate:

So $\overline{BC}$ is vertical.

A horizontal line and a vertical line are perpendicular. Therefore, the angle at $B$ is a right angle.

So, $\triangle ABC$ is a right triangle.

Step 2: Find the base and height

The length of the horizontal segment $\overline{AB}$ is the change in the $x$-coordinates:

The length of the vertical segment $\overline{BC}$ is the change in the $y$-coordinates:

Thus, the base is $4$ units and the height is $3$ units.

Step 3: Find the area

Use the triangle area formula:

Substitute the values:

Therefore, the triangle is a right triangle, and its area is

When working with coordinates, remember that equal $y$-coordinates indicate a horizontal segment, equal $x$-coordinates indicate a vertical segment, and lengths are found using nonnegative changes such as absolute values.