If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is

The correct answer is 60.

Given the properties of a rhombus, let the side length be $a$ and the lengths of its diagonals be $d_1$ and $d_2$.

We are given the following values:

Side length $a=36$ cm

Area $A=396$ sq. cm

The area of a rhombus can be expressed in terms of its diagonals as:

$\text{Area}=\frac{1}{2}\times d_1\times d_2$

Substitute the given area into the formula to find the product of the diagonals:

$396=\frac{1}{2}\times d_1\times d_2$

$d_1{d}_2=396\times 2=792$

Another important property of a rhombus is the relationship between its side length and diagonals. Because the diagonals of a rhombus bisect each other at right angles, they form four right-angled triangles. Using the Pythagorean theorem, the sum of the squares of the diagonals is equal to four times the square of the side length:

$d_1^2+d_2^2=4a^2$

Substitute the given side length ($a=36$) into this equation:

$d_1^2+d_2^2=4{\left(36\right)}^2$

$d_1^2+d_2^2=4\times 1296$

$d_1^2+d_2^2=5184$

We need to find the absolute difference between the lengths of the diagonals, which is $|d_1-d_2|$. We can use the following algebraic identity:

${(d_1-d_2)}^2=d_1^2+d_2^2-2d_1{d}_2$

Now, substitute the values we have already found for the sum of squares and the product of the diagonals:

${(d_1-d_2)}^2=5184-2\left(792\right)$

${(d_1-d_2)}^2=5184-1584$

${(d_1-d_2)}^2=3600$

Taking the square root of both sides gives the absolute difference:

$|d_1-d_2|=\sqrt{3600}$

$|d_1-d_2|=60$

Therefore, the absolute difference between the lengths of the diagonals is 60 cm.