The diagonals of the rhombus are 3: 4, and the area is 54 cm 2. Find the lengths of the diagonals of the rhombus.

Given a rhombus ABCD. AC and BD are its diagonals. It is known that they are related as 3: 4, that is, AC = 3x, and BD = 4x, where x is the proportionality coefficient.

By condition, the area of the rhombus is 54 cm ^ 2.

And since the diagonals of the rhombus are perpendicular, the area of the rhombus is found by the formula:

S = 1/2 * d1 * d2 = 1/2 * AC * BD = 1/2 * 3x * 4x = 6x ^ 2.

Substitute the area value and find the coefficient x:

54 = 6x ^ 2;

x ^ 2 = 54/6;

x ^ 2 = 9;

x = 3.

Now let’s find the diagonals:

AC = 3x = 3 * 3 = 9cm

BD = 4x = 4 * 3 = 12 cm.

Answer: the lengths of the diagonals of the rhombus are 9 cm and 12 cm.