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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:
What is the area of the quadrilateral in the coordinate system?
  1. 2
  2. 4
  3. 6
  4. 8
  5. 11

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.
The answer is (D).


What is the perimeter of Triangle ABC in the figure?

Point A has coordinates (0, 4), point B has coordinates (3, 0), and point C has coordinates (5, 1).


Using the distance formula to calculate the distances between points A and B, A and C, and B and C yields


Adding these lengths gives the perimeter of Triangle ABC:


The answer is (A).

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