Question

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:

Hint:

For any rhombus with side length \(a\) and diagonals \(d_1, d_2\), use: \[ d_1^2 + d_2^2 = 4a^2, \qquad \text{and} \qquad \text{Area} = \frac12 d_1 d_2. \] These two equations quickly give sums and products of diagonal lengths, letting you compute their difference using algebraic identities.

Answer 1

Correct Answer: 60

Step 1: Using \(\text{Area} = \frac{d_1 d_2}{2}\), we get \(d_1 d_2 = 2\times396 = 792\).
Step 2: Using the side-length relation \(\left(\frac{d_1}{2}\right)^2+\left(\frac{d_2}{2}\right)^2 = 36^2\), we get \(d_1^2+d_2^2 = 4\times1296 = 5184\).
Step 3: \((d_1-d_2)^2 = d_1^2+d_2^2-2d_1d_2 = 5184-1584=3600\), so \(d_1-d_2=60\).
\[ \boxed{60 \text{ cm}} \]