Question

If rectangle ABCD ~ rectangle EFGH, and the area of ABCD is 64 cm^2 and the area of EFGH is 49 cm^2, what is the scale factor of ABCD to EFGH?

Answer 1

7 / 8

Explanation:

The scale factor between two-dimensional shapes is the ratio of one of their corresponding side lengths.

For example, to go from a square whose side lengths are 6 cm to one whose side lengths are 3 cm, the scale factor would be 1/2, since 3 cm / 6 cm = 1/2.

The ratio of the areas of similar rectangles is the (scale factor)², since the scale factor is multiplied by both sides of the triangle, and therefore, there are two instances of it in the area equation for the dilated shape.

For example, to go from a square whose area is 36 cm² to one whose side lengths are 9 cm², the scale factor would be 1/2, since 9 cm² / 36 cm² = (1/2)².

Applying this concept to the problem at hand:

(scale factor)² = (Area of ABCD) / (Area of EFGH)

↓ plugging in the given values

(scale factor)² = 49 cm² / 64 cm²

↓ take the square root of both sides

scale factor =
`√(49)` /
`√(64)`

↓ simplifying the square roots

scale factor = 7 / 8