Question

In a square, lengths of the diagonals are (4k+6) cm and (7k -3) cm. The area of the square (in \(\text{cm}^2\)) is:

• A. 144
• B. 162
• C. 169
• D. 172

Hint:

The area of a square can be expressed using either its side \( a \) or its diagonal \( d \).
While the basic formula is \( \text{Area} = a^2 \), using the diagonal formula \( \text{Area} = \frac{d^2}{2} \) directly avoids the extra step of finding the side length \( a = \frac{d}{\sqrt{2}} \), saving time.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A square is a regular quadrilateral, which means all its sides are equal and both of its diagonals are equal in length. By setting the expressions for the two diagonals equal to each other, we can solve for the unknown variable $k$ and find the actual length of the diagonal.
Step 2: Key Formula or Approach:
- Property of a square: \[ d_1 = d_2 \] - Area of a square given its diagonal ($d$): \[ \text{Area} = \frac{1}{2} \times d^2 \]
Step 3: Detailed Explanation:
Equate the two diagonal expressions to solve for $k$: \[ 7k - 3 = 4k + 6 \] Subtract $4k$ from both sides: \[ 3k - 3 = 6 \] Add 3 to both sides: \[ 3k = 9 \] \[ k = 3 \] Substitute $k = 3$ back into one of the diagonal expressions to find the length ($d$): \[ d = 4(3) + 6 = 12 + 6 = 18 \text{ cm} \] Now, calculate the area of the square using the diagonal: \[ \text{Area} = \frac{1}{2} \times 18^2 \] \[ \text{Area} = \frac{1}{2} \times 324 = 162 \text{ cm}^2 \]
Step 4: Final Answer:
The area of the square is 162 $\text{cm}^2$.