McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
2. Areas of Trapezoids, Rhombi, and Kites
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Exercise 31 Page 795

The perimeter of a trapezoid is the sum of its four side lengths. The area is one half the product of the height and the sum of the bases.

Perimeter:
Area:

Practice makes perfect

For the given polygon we will find its perimeter and its area one at a time. Let's do it!

Perimeter

The perimeter of a polygon is found by adding all of its side lengths. We know the lengths of two sides of our figure, but we are missing the other two. Since the polygon has exactly one pair of parallel sides, it is a trapezoid. Also, since the nonparallel sides are congruent, the trapezoid is an isosceles trapezoid.

Let's draw another height to construct a second right triangle on the right-hand side of the diagram.

The right triangles have congruent hypotenuses and one pair of congruent legs. Therefore, by the Hypotenuse Leg Theorem the triangles are congruent. Let be the length of their base.

We can find the value of by using the Segment Addition Postulate.
Let's solve this equation.
Solve for
The length of the base of each right triangle is inches. Let's now pay close attention to just one of them.
We have a right triangle with an acute angle that measures and whose opposite side length is inches. To find the length of the hypotenuse — which is also a side of the trapezoid — we will use the sine ratio.
If we let be the length of the hypotenuse of the triangle, we can substitute the corresponding values in the above equation to find its length.
Solve for
The hypotenuse of the right triangle has a length of inches. Note that the opposite side is congruent. Therefore, its length is also inches.
We have all of the information we need to calculate the perimeter of the polygon.
Finally, to convert inches into feet, we will use the fact that inches is equal to foot. We have to multiply our answer by the conversion factor

Area

The area of a trapezoid is one half the product of its base and the sum of its bases. In the given diagram, we can see that the lengths of the bases are and inches. Let's use a right triangle again to find the height

The length of the hypotenuse is inches and the length of one of the legs inches. The length of the missing leg is also the height of the trapezoid. We can find it by substituting and into the Pythagorean Theorem. Let's do it!
Solve for
Recall that represents a side of a triangle and the height of the trapezoid. Therefore, we only kept the principal root when solving the equation because must be positive. The height of the trapezoid is inches.
Having the height and the lengths of the bases, we can substitute them into the formula for the area of a trapezoid.
Evaluate right-hand side
The area of the trapezoid is To express it in square feet, we have to multiply it by twice.
Finally, let's simplify this expression.
Simplify