9.1.1 Developing Formulas for Triangles and Quadrilaterals Key Objectives • Develop and apply the formulas for the areas of triangles and special quadrilaterals. • Solve problems involving perimeters and areas of triangles and special quadrilaterals. Formulas • The area of a parallelogram with base b and height h is A = bh . • The area of a triangle with base b and height h is A bh . =
1 2
b b h ) 1 2 +
(
1 2
• The area of a trapezoid with bases b 1 and b 2 and height h is A b b h A ( ) , or 1 2 = +
.
=
2
• The area of a rhombus or kite with diagonals d 1 and d 2 is A d d 1 2 . 1 2 = Theorems, Postulates, Corollaries, and Properties • Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. Example 1 Finding Measurements of Parallelograms The area of a parallelogram with base b and height h is A = bh .
This is the same as the formula for the area of a rectangle. One way to think about why they are the same is to overlay the two figures and see that the rectangle can be constructed by removing a triangle from one end of the parallelogram and adding a congruent triangle to the other end of the parallelogram. The area of a parallelogram is determined in this example. The length of the base, the length of the sides, and the length of the overhang are given. To calculate the height, use the Pythagorean Theorem with the parallelogram side as the hypotenuse and the overhang as a leg. The height of the parallelogram is 3 cm. The area of the parallelogram is A = bh = 12(3) = 36 cm 2 .
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