Question

In a square, the lengths of the diagonals are $(4k + 6)$ cm and $(7k - 3)$ cm. What is the area of the square (in cm²)?

• A. \(144\)
• B. \(169\)
• C. \(162\)
• D. \(172\)

Answer 1

The Correct Option is C

To find the area of the square, we first need to determine the side length of the square using the diagonals. Since the diagonals of a square are equal, we equate them: 

$(4k + 6) = (7k - 3)$

Solving for $k$:

1. Rearrange the equation: $4k + 6 = 7k - 3$

2. Subtract $4k$ from both sides: $6 = 3k - 3$

3. Add $3$ to both sides: $9 = 3k$

4. Divide by $3$: $k = 3$

Now substitute the value of $k$ back into one of the equations for the diagonal:

Length of diagonal = $4(3) + 6 = 18$ cm or $7(3) - 3 = 18$ cm

In a square, the relationship between the side length $s$ and the diagonal $d$ is given by: $d = s\sqrt{2}$

So, $18 = s\sqrt{2}$

Solving for $s$: $s = \frac{18}{\sqrt{2}} = 18 \times \frac{\sqrt{2}}{2} = 9\sqrt{2}$ cm

The area of the square is $s^2$:

Area = $(9\sqrt{2})^2 = 81 \times 2 = 162$ cm²

The correct answer, therefore, is 162 cm².

Answer 2

In a square, the diagonals are equal in length. So, \(4k + 6 = 7k - 3\).

\(3k = 9\)

\(k = 3\)

The length of a diagonal is \(4(3) + 6 = 12 + 6 = 18\) cm.

The area of a square is half the square of its diagonal: Area = \(\frac{1}{2} \times d^2\), where d is the diagonal

Area = \(\frac{1}{2} \times 18^2 = \frac{1}{2} \times 324 = 162\) cm²