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

Question

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

Accepted Answer

1. **Equate the diagonals:** Since both expressions represent the diagonal of the square, equate them:  
   $$
   4k + 6 = 7k - 3
   $$

2. **Solve for $$ k $$:**  
   Rearrange and simplify:  
   $$
   7k - 4k = 6 + 3 \quad \Rightarrow \quad 3k = 9 \quad \Rightarrow \quad k = 3
   $$

3. **Find the diagonal length:** Substitute $$ k = 3 $$ into either expression for the diagonal:  
   - $$ 4k + 6 = 4(3) + 6 = 18 \, 	ext{cm} $$.

4. **Calculate the area of the square:** The formula for the area of a square using its diagonal is:  
   $$
   	ext{Area} = \frac{	ext{Diagonal}^2}{2}
   $$  
   Substituting the diagonal length:  
   $$
   	ext{Area} = \frac{18^2}{2} = \frac{324}{2} = 162 \, 	ext{cm}^2
   $$

Thus, the area of the square is **162 cm$$^2$$**, making option **(2)** correct.

Answer

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