c Draw the new rectangle and compare to the original.
D
d Use the formulas for perimeter and area.
A
a 14ft
B
b 12ft^2
C
cPerimeter: Doubles Area: Quadruples
Explanation: See solution.
D
dPerimeter: Halved Area: 14 the original area
Explanation: See solution.
Practice makes perfect
a The markers on the sides indicate that this rectangle has two pairs of congruent sides. The length, l=4 feet, and width, w=3 feet, of the rectangle is also given on the figure.
We can use the formula P=2l+2w to find the perimeter of this rectangle.
c The diagram below shows the rectangle with the doubled sides, along with the original rectangle.
Counting the line segments with the same markers on the diagram, we can see that the perimeter doubles. To justify this observation more formally, lets use the formula for the perimeter.
P_(original)=2l+2w
P_(new)=2(2l)+2(2w)We can express the new perimeter in terms of the original perimeter.
When we double the sides of a rectangle, the perimeter also doubles.
To see how the area changes, let's look at the relationship between the new and original rectangles a bit differently.
We can see that four congruent copies of the original rectangle make up the new rectangle, so the area quadruples. To justify this observation more formally, let's use the formula for the area.
A_(original)=l w
A_(new)=2l* 2w
We can express the new area in terms of the original area.
When we double the sides of a rectangle, the area quadruples.
d We just examined the effect on area and perimeter if the lengths are doubled, and now we will look at the results if they are halved. Let's use the formula for the perimeter, and create a new one to show the halving.
P_(original)=2l+2w
P_(new)=2(1/2l)+2(1/2w)
We can express the new perimeter in terms of the original perimeter.
When the sides of a rectangle are halved, the perimeter is also halved.
To see what happens to the area, let's use the formula for the area and our new measurements.
A_(original)=l w
A_(new)=1/2l* 1/2w
We can express the new area in terms of the original area.
