Question

In a parallelogram shown below, \( a^2 + b^2 = ? \)

• A. \( \frac{d_1^2 - d_2^2}{4} \)
• B. \( \frac{d_1^2 - d_2^2}{2} \)
• C. \( \frac{d_1^2 + d_2^2}{2} \)
• D. \( \frac{d_1^2 + d_2^2}{4} \)

Hint:

The sum of the squares of the sides of a parallelogram is related to the squares of the diagonals by \( a^2 + b^2 = \frac{d_1^2 + d_2^2}{2} \).

Answer 1

The Correct Option is C

Step 1: Understanding the problem.
In the given parallelogram, we need to find the sum of the squares of the sides \( a \) and \( b \), i.e., \( a^2 + b^2 \), using the diagonals \( d_1 \) and \( d_2 \). Step 2: Relation between sides and diagonals.
For any parallelogram, the relation between the sides \( a \), \( b \) and the diagonals \( d_1 \), \( d_2 \) is given by the following formula: \[ a^2 + b^2 = \frac{d_1^2 + d_2^2}{2} \] Step 3: Conclusion.
Thus, the sum of the squares of the sides of the parallelogram is \( \frac{d_1^2 + d_2^2}{2} \), which corresponds to option (C).