The areas of the similar rhombi are 300sq. units and 768sq. units The larger diameter of the smaller rhombus is congruent to the side of the bigger rhombus. Find the perimeters of the rhombi.

smaller: 50√2 units larger: 80√2 units

Step-by-step explanation:

The ratio of side lengths of the rhombi is the square root of the ratio of their areas:

scale factor = √ (768/300) = √ (64/25) = 8/5

The long diagonal of the smaller rhombus is the same length as the side of the larger rhombus. This means the length of the long diagonal of the smaller rhombus is 8/5 times the length of its side. So the long semi-diagonal is 4/5 of the side length, and the short semi-diagonal is ...

√ (1 - (4/5) ²) = √ (9/25) = 3/5

of the side length. The area of the rhombus in terms of its side length s is ...

A = 2 (3/5s) (4/5s) = (24/25) s²

and the side length in terms of area is ...

s = √ (25A/24) = (5/2) √ (A/6)

The perimeter is 4 times the side length, so is ...

P = 4s = 4 (5/2) √ (A/6) = 10√ (A/6)

For the smaller rhombus, the perimeter is 10√ (300/6) = 50√2 units.

For the larger rhombus, the perimeter is (8/5) (50√2) = 80√2 units.