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: Understanding the Question:
We are given the side length and area of a rhombus and asked to find the absolute difference between its diagonals, \(|d_1 - d_2|\).
Step 2: Key Formula or Approach:
We need two key properties of a rhombus:
1. The area (A) in terms of its diagonals (\(d_1, d_2\)): \(A = \frac{1}{2} d_1 d_2\).
2. The relationship between the side (s) and the diagonals: \(4s^2 = d_1^2 + d_2^2\).
We will also use the algebraic identity: \((x - y)^2 = x^2 + y^2 - 2xy\).
Step 3: Detailed Explanation:
Given data: \(s = 36\) cm and \(A = 396\) cm\(^2\).
Part A: Find the product of the diagonals (\(d_1 d_2\))
Using the area formula:
\[ 396 = \frac{1}{2} d_1 d_2 \] \[ d_1 d_2 = 2 \times 396 = 792 \] Part B: Find the sum of the squares of the diagonals (\(d_1^2 + d_2^2\))
Using the side-diagonal relationship:
\[ d_1^2 + d_2^2 = 4s^2 = 4 \times (36)^2 = 4 \times 1296 = 5184 \] Part C: Find the difference between the diagonals
We want to find \(|d_1 - d_2|\). We can find \((d_1 - d_2)^2\) first using the algebraic identity:
\[ (d_1 - d_2)^2 = (d_1^2 + d_2^2) - 2(d_1 d_2) \] Substitute the values we found:
\[ (d_1 - d_2)^2 = 5184 - 2(792) = 5184 - 1584 = 3600 \] Now, take the square root of both sides:
\[ |d_1 - d_2| = \sqrt{3600} = 60 \] Step 4: Final Answer:
The absolute value of the difference between the lengths of the diagonals is 60 cm.