# Areas and Perimeters

Often, you will be given a geometric figure drawn on a coordinate system and will be asked to find its area or perimeter. In these problems, use the properties of the coordinate system to deduce the dimensions of the figure and then calculate the area or perimeter. For complicated figures, you may need to divide the figure into simpler forms, such as squares and triangles.

A couple examples will illustrate:

Example
What is the area of the quadrilateral in the coordinate system?
1. 2
2. 4
3. 6
4. 8
5. 11
Solution
If the quadrilateral is divided horizontally through the line y = 2, two congruent triangles are formed.

As the figure shows, the top triangle has height 2 and base 4.

Hence, its area is  .

The area of the bottom triangle is the same, so the area of the quadrilateral is 4 + 4 = 8.

Example
What is the perimeter of Triangle ABC in the figure?
Solution
Point A has coordinates (0, 4), point B has coordinates (3, 0), and point C has coordinates (5, 1).

Adding these lengths gives the perimeter of Triangle ABC:Using the distance formula to calculate the distances between points A and B, A and C, and B and C yields