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 calculate the area of the square, we first determine its side length. This is achieved by using the given diagonals, which are equal in length. We set up the equation:

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

The value of $k$ is found by solving this equation:

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

2. Isolate the $k$ term: $6 = 3k - 3$

3. Isolate the constant term: $9 = 3k$

4. Solve for $k$: $k = 3$

Substitute $k=3$ into either diagonal equation to find the diagonal length:

Diagonal length = $4(3) + 6 = 18$ cm or $7(3) - 3 = 18$ cm

For a square, the relationship between the side length ($s$) and the diagonal ($d$) is $d = s\sqrt{2}$. Using the calculated diagonal length:

$18 = s\sqrt{2}$

Solving for $s$ yields:

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

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

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

The area of the square is 162 cm².