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 Question:
In a square, both diagonals are of equal length.
We are given two algebraic expressions representing the lengths of these diagonals: \( (4k+6)\text{ cm} \) and \( (7k-3)\text{ cm} \).
We need to determine the area of the square using these given expressions.

Step 2: Key Formula or Approach:
Since the diagonals of a square are always equal, we equate the two expressions to find the value of the unknown variable \( k \):
\[ 4k + 6 = 7k - 3 \]
Once the value of \( k \) is found, we substitute it back to compute the actual length of the diagonal \( d \).
The area of a square in terms of its diagonal \( d \) is given by:
\[ \text{Area} = \frac{1}{2} \times d^2 \]

Step 3: Detailed Explanation:
1. Let us denote the two diagonals of the square as \( d_1 \) and \( d_2 \).
2. Since the diagonals of any square are congruent, we have:
\[ d_1 = d_2 \]
3. Substituting the given algebraic expressions:
\[ 4k + 6 = 7k - 3 \]
4. To solve for \( k \), we rearrange the terms by shifting the variable term \( 4k \) to the right-hand side and the constant term \( -3 \) to the left-hand side:
\[ 6 + 3 = 7k - 4k \]
5. Simplifying both sides yields:
\[ 9 = 3k \]
6. Dividing by 3, we find the value of \( k \):
\[ k = 3 \]
7. Now, we substitute this value of \( k \) back into the expression for either diagonal to find its length:
\[ d = 4k + 6 = 4(3) + 6 = 12 + 6 = 18\text{ cm} \]
8. We can also verify this value using the second expression:
\[ d = 7(3) - 3 = 21 - 3 = 18\text{ cm} \]
The diagonal length is consistently calculated as \( 18\text{ cm} \).
9. Next, we compute the area of the square using the diagonal length:
\[ \text{Area} = \frac{1}{2} \times d^2 = \frac{1}{2} \times (18)^2 \]
10. Calculating the square of 18 gives 324:
\[ \text{Area} = \frac{1}{2} \times 324 = 162\text{ cm}^2 \]

Step 4: Final Answer:
The area of the square is \( 162\text{ cm}^2 \), which corresponds to option (B).