9.2.1 Perimeter and Area in the Coordinate Plane Key Objectives • Find the perimeters and areas of figures in a coordinate plane. Example 1 Estimating Areas of Irregular Shapes in the Coordinate Plane
The area of an irregular shape in the coordinate plane is estimated using two methods. In the first method, the irregular shape is approximated by filling it with smaller regular figures, such as triangles and rectangles, whose areas can be calculated. The area of the irregular shape is approximated by the area of the composite shape. This method yields an area estimate of 38.5 square units. In the second method, the number of full square units enclosed by the irregular shape is counted. Then the number of square units with a portion lying within the irregular shape is counted. The area of the irregular shape is estimated as the number of full square units plus one-half the number of partial square units. This method yields an estimate of 38 square units.
Example 2 Finding Perimeter and Area in the Coordinate Plane
The type of polygon drawn in the coordinate plane and its perimeter and area are determined in this example. The coordinates of the vertices of the polygon are given. The lengths of the sides of the polygon are determined using the Distance Formula. All four sides are congruent, so the figure is a rhombus. The perimeter of the rhombus is 4 10. The slopes of the sides can be determined from the formula for slope and the coordinates of the vertices. None of the products of the slopes of adjacent sides are − 1, so none of the sides are perpendicular and this is not a square. The area is one-half times the product of the lengths of the diagonals. The lengths of the diagonals can be calculated using the Distance Formula and the coordinates of opposite vertices. The area is found to be 8 square units.
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