Question

Two figures are similar with a scale factor of 5/2.

a. What is the ratio of corresponding lengths?
b. What is the ratio of their perimeters?
c. What is the ratio of their areas?
d. What is the ratio of their volumes?
e. What is the ratio of corresponding angle measures?

Answer 1

For two similar figures with a scale factor of 5/2, the ratio of corresponding lengths and perimeters is 5/2, the ratio of areas is 25/4, the ratio of volumes is 125/8, and the ratio of corresponding angle measures is 1, as angles remain unchanged.

Step-by-step explanation:

When two figures are similar with a scale factor of 5/2, the following ratios are established:

a. Corresponding Lengths: The ratio of corresponding lengths is directly the scale factor, which is 5/2.

b. Perimeters: The ratio of their perimeters is also the scale factor, as the perimeter of a shape is a one-dimensional measure, so it too is 5/2.

c. Areas: The ratio of their areas is the square of the scale factor. Therefore, we calculate this as (5/2)^2 which is 25/4.

d. Volumes: The ratio of their volumes is the cube of the scale factor. Hence, it is (5/2)^3 which equals 125/8.

e. Corresponding Angle Measures: Since angles are not affected by size changes in similar figures, the ratio of corresponding angle measures is always 1, meaning they are equal.

Answer 2

If we have two figures, F and F'

Such that if we start with F, and dilate it with a scale factor K, we get F'.

We will have:

All the measures of F', are K times the correspondent measures of F.

Then if F has s₁, s₂, ..., sₙ sides, the sides of F' will be:

K*s₁, K*s₂, ..., K*sₙ

The ratio between correspondent sides will be equal to K

The ratio between perimeters will also be equal to K (because the perimeter is the sum of all the sides of each figure, so we can just take K as a common factor)

In the case of the area, because we usually multiply a measure by another, a factor K^2 will appear, and the quotient between the areas is K^2

And finally, for the volumes, the ratio will be K^3

a) The ratio of corresponding lengths is K, in this case is 5/2

b) The ratio of the perimeters is K, in this case is 5/2

c) The ratio of the areas is K^2, in this case is (5/2)^2 = 25/4

d) The ratio of the areas is K^3, in this case is (5/2)^3 = 125/8

e) Two figures are similar if the figures have the same shape, then the corresponding angles are exactly the same, then the ratio of corresponding angle measures is 1.