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

Entered answer:

✅ Correct Answer: 60

Let $d_1$ and $d_2$ be the two diagonals of the rhombus.

The diagonals of a rhombus bisect each other at 90 degrees, so each side forms a right triangle with half-diagonals:

$\text{side}^2 = \left(\dfrac{d_1}{2}\right)^2 + \left(\dfrac{d_2}{2}\right)^2$

Using the area formula for a rhombus:

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

$d_1 \cdot d_2 = 792 \quad \text{...(i)}$

Using the side length and the right triangle property:

$36^2 = \dfrac{d_1^2}{4} + \dfrac{d_2^2}{4}$

$1296 = \dfrac{d_1^2 + d_2^2}{4}$

$d_1^2 + d_2^2 = 5184 \quad \text{...(ii)}$

Using the identity $(d_1 - d_2)^2 = (d_1^2 + d_2^2) - 2(d_1 \cdot d_2)$ with (i) and (ii):

$(d_1 - d_2)^2 = 5184 - 2(792)$

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

$(d_1 - d_2)^2 = 3600$

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

$|d_1 - d_2| = 60$

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

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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

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