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

Let the diagonals of the rhombus be \(d_1\) and \(d_2\). The length of each side of the rhombus is given as \(a = 36\) cm. Step 1: Apply the formula for the area of a rhombus. The area of a rhombus in terms of its diagonals is \[ \text{Area} = \frac{1}{2} d_1 d_2. \] Given that the area is \(396\ \text{cm}^2\), we have \[ \frac{1}{2} d_1 d_2 = 396, \] which gives \[ d_1 d_2 = 792. \] Step 2: Use the relationship between diagonals and side. In a rhombus, the diagonals bisect each other at right angles. Using the Pythagorean theorem, \[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2. \] Multiplying both sides by 4, \[ d_1^2 + d_2^2 = 4a^2 = 4(36^2) = 5184. \] Step 3: Find the difference between the diagonals. Using the identity \[ (d_1 - d_2)^2 = d_1^2 + d_2^2 - 2d_1 d_2, \] and substituting the known values, \[ (d_1 - d_2)^2 = 5184 - 2(792) = 5184 - 1584 = 3600. \] Taking the square root, \[ |d_1 - d_2| = \sqrt{3600} = 60. \] Hence, the required difference between the diagonals is \(60\).

Answer 2

Let the diagonals of the rhombus be \(d_1\) and \(d_2\). Side length of the rhombus is \(a = 36\) cm. Step 1: Use the area formula. \[ \text{Area} = \frac{1}{2} d_1 d_2 = 396 \] \[ d_1 d_2 = 792. \]
Step 2: Use the diagonal–side relationship. Diagonals of a rhombus bisect each other at right angles, so by Pythagoras: \[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2. \] Multiply by 4: \[ d_1^2 + d_2^2 = 4a^2 = 4(36^2) = 5184. \]
Step 3: Compute \(|d_1 - d_2|\). Use the identity: \[ (d_1 - d_2)^2 = d_1^2 + d_2^2 - 2 d_1 d_2. \] Substitute known values: \[ (d_1 - d_2)^2 = 5184 - 2(792) = 5184 - 1584 = 3600. \] \[ |d_1 - d_2| = \sqrt{3600} = 60. \] Thus, the required difference between the diagonals is: \[ \boxed{60}. \]