9.2.2 Effects of Changing Dimensions Proportionally
Key Objectives • Describe the effect on perimeter and area when one or more dimensions of a figure are changed. • Apply the relationship between perimeter and area in problem solving.
Changing dimensions proportionally means changing all dimensions of a figure by the same factor. This type of change has the effect of changing the perimeter or circumference of a figure by the same factor. However, the area of the figure changes by the square of the factor. This is expected because the area is a
function of the square of the linear dimensions of a figure. Example 1 Effects of Changing One Dimension
The effect of doubling the height of a triangle on the area of the triangle is determined in this example. To calculate the area of a triangle with height 7 inches and base 10 inches, substitute the values into the formula for area. A = 1/2(10)(7) = 35 in 2 . To calculate the area of a triangle with height 14 inches and base 10 inches, substitute the values into the formula for area. A = 1/2(10)(14) = 70 in 2 . When the height of the triangle is doubled, the area of the triangle also doubles. Note that this is not a proportional change in dimensions; only one dimension is changed. The effect of halving the base length of a parallelogram on the area of the parallelogram is determined in this example. To calculate the area of a parallelogram with a base length of 3 units and height of 2 units, substitute the values into the formula for area. A = (3)(2) = 6 units 2 . To calculate the area of a parallelogram with a base length of (1/2)3 units and height of 2 units, substitute the values into the formula for area. A = (1/2)(3)(2) = 3 units 2 . When the length of the base of the parallelogram is halved, the area of the parallelogram also halves.
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