Question

If the length of each side of a rhombus is 36 cm, and the area of the rhombus is 396 cm\(^2\), then what is the absolute value of the difference between the lengths of its diagonals?

Hint:

For rhombus problems, the two key formulas relating area, side, and diagonals are essential. Combining them with algebraic identities like \((x \pm y)^2\) is a common solution pattern.

Answer 1

Correct Answer: 60

Step 1: Treat \(d_1,d_2\) as the two roots of a quadratic \(t^2-St+P=0\), where \(S=d_1+d_2\) and \(P=d_1d_2\). From the area, \(P=792\).
Step 2: From the side length, \(d_1^2+d_2^2=4(36)^2=5184\), and since \(d_1^2+d_2^2=S^2-2P\), we get \(S^2=5184+2(792)=6768\).
Step 3: For a quadratic with roots \(d_1,d_2\), the discriminant equals \((d_1-d_2)^2=S^2-4P=6768-3168=3600\).
Final Answer: \[ |d_1-d_2|=\sqrt{3600}=\boxed{60\text{ cm}} \]