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.
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