write a ratio comparing widths, lengths, and the perimeter of a rectangle 6 inches by 9 inches and a rectangle 8 inches by 12 inches. Show whether these are are equivalent or not.
Rectangle1: W1 = 6in, L1 = 9in.
P1 = 2W1 + 2L1 = 12 + 18 = 30 in.
Retangle2: W2 = 8in, L2 = 12in.
P2 = 2W + 2L2 = 16 + 24 = 40 in.
W1:W2 = 6:8 = 3:4
L1:L2 = 9:12 = 3:4
P1:P2 = 30:40 = 3:4.
All of the ratios are equal. Therefore,
the rectangles are similar.
To write the ratio comparing the widths, lengths, and perimeters of the given rectangles, we need to determine the values for each.
Rectangle 1:
Width = 6 inches
Length = 9 inches
Perimeter = 2(Width + Length) = 2(6 + 9) = 2(15) = 30 inches
Rectangle 2:
Width = 8 inches
Length = 12 inches
Perimeter = 2(Width + Length) = 2(8 + 12) = 2(20) = 40 inches
Now, let's write the ratio for each measurement:
1. Width ratio:
Width of Rectangle 1 : Width of Rectangle 2
6 inches : 8 inches
2. Length ratio:
Length of Rectangle 1 : Length of Rectangle 2
9 inches : 12 inches
3. Perimeter ratio:
Perimeter of Rectangle 1 : Perimeter of Rectangle 2
30 inches : 40 inches
To determine if the ratios are equivalent or not, we need to simplify them.
1. Simplifying the width ratio:
We can divide each value in the ratio by their greatest common divisor (GCD), which is 2.
6 inches ÷ 2 = 3 inches
8 inches ÷ 2 = 4 inches
So, the simplified width ratio is:
3 inches : 4 inches
2. Simplifying the length ratio:
Again, we find the GCD, which is 3, in this case.
9 inches ÷ 3 = 3 inches
12 inches ÷ 3 = 4 inches
Thus, the simplified length ratio is:
3 inches : 4 inches
3. Simplifying the perimeter ratio:
Once more, we determine the GCD, which is 10 here.
30 inches ÷ 10 = 3 inches
40 inches ÷ 10 = 4 inches
Hence, the simplified perimeter ratio is:
3 inches : 4 inches
Comparing the simplified ratios of the widths, lengths, and perimeters, we can observe that they are all equivalent. Therefore, the ratios are equal, and the rectangles have a proportional relationship.