9.2.1 Perimeter and Area in the Coordinate Plane (continued)
Example 3 Finding Areas in the Coordinate Plane by Subtracting
The area of a polygon in the coordinate plane is found in this example. The coordinates of the vertices of the polygon are given. To calculate the area of the polygon, use the indirect method of subtraction. The rectangle that encloses the polygon has side lengths of 6 units and 5 units. Its area is (6)(5) = 30 square units. Outside the polygon, but within the rectangle, are four triangles whose areas can be easily calculated because they are right triangles and their bases and heights are simply leg lengths. The sum of the area of the four triangles is 16 square units. The area of the polygon is the area of the rectangle minus the sum of the area of the four triangles, or 14 square units. Two composite figures are formed using four shapes. The composite figures appear to have different areas. The area of the shapes themselves is 11.5 square units. The area of the left composite figure, estimated as one-half times base times height, is 11 square units. The area of the right composite figure, estimated as one-half times base times height, is 12 square units. The difference appears to be a paradox. The paradox is resolved by recognizing that both composite figures are not triangles, but quadrilaterals. The slope of the hypotenuse in the large triangle is 2/5, while the slope of the hypotenuse of the small triangle is 1/3. The slopes are not the same, so the hypotenuses do not form a straight line in the composite figures. The true area of both figures is the area obtained by summing the areas of the shapes that make them up, or 11.5 square units.
Example 4 Problem-Solving Application
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