Bchapter 3

example #1 Areas of Polygons in the Coordinate Plane

It's also possible to find the area of a polygon if you know the coordinates of its vertices, or the endpoints of the segments that are needed to find the area, because then you can use the distance formula to find the necessary segment lengths. GO TO IMAGE

example #1 Perimeters and Areas of Polygons

We can find the perimeter of a parallelogram by finding the sum of its side lengths. The opposite sides of a parallelogram are congruent, which means they have equal lengths. Using the image below, you can see that the perimeter would be 2 side lengths of a plus 2 base lengths of b. Therefore, the perimeter of a parallelogram is twice the sum of a plus b, when b is the base length and a is the adjacent side length. On the other hand, the area of a parallelogram is the product of its base and its height.

Perimeter of Polygons in the Coordinate Plane

You can also find the perimeter of a polygon if you're given only the coordinates of a polygon's vertices. In that case, you can use the distance formula to find all necessary lengths, and then you can determine the perimeter from those lengths. GO TO IMAGE

Example #2 Now, let's use consecutive vertices,

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Example #5 Our last polygon is the triangle. The perimeter of any triangle is the sum of its side lengths, and the area is half the product of its base and height.

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example #1 to find the perimeter of a rectangle, ABCD, with vertices A(−3,−2), B(0,4), C(4,2), and D(1,−4), you do not need to find every side length. It is sufficient to find just the length and width of the rectangle.

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example #2 a rectangle, a special type of parallelogram, is a quadrilateral with four right angles. A rectangle's perimeter and area can be found in a way that is similar to finding the perimeter and area of a parallelogram.

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example #2 to find the area of this trapezoid, we need the lengths of the bases, which are the parallel sides, and the height. First, we'll find the lengths of the bases, AB and CD

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example #3 A square is also a parallelogram with four right angles and four congruent sides. The perimeter and the area of a square are found in the same way as the perimeter and area of a rectangle, but they are simpler to calculate in a square because the sides all have the same length.

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example #3 Next, we'll find the height, or the length of AE, which has endpoints at A(−1,4) and E(−1,−2)

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example #3 Then, we can use these segment lengths as the length and width of the rectangle to find the perimeter of rectangle ABCD.

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example #4 The perimeter of a trapezoid is the sum of its side lengths, and the area is half the product of its height and the sum of its bases.

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example #4 Then, we can use the formula for the area of a trapezoid to find the area of ABCD.

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