The diagonal of a square is 12 cm. What is the length (in cm) of each side of the square?

Introduction / Context:
Squares are special quadrilaterals in which all sides are equal and each interior angle is a right angle. The diagonal of a square connects opposite vertices and forms a right triangle with two sides of the square as the legs. Understanding the relationship between the side length and the diagonal of a square is a standard topic in geometry and appears frequently in aptitude exams.

Given Data / Assumptions:

• The figure is a square.
• The length of its diagonal is 12 cm.
• We must determine the length of each side of the square.

Concept / Approach:
If we denote the side length of the square by a, then the diagonal forms a right triangle with legs a and a and hypotenuse equal to the diagonal. By the Pythagorean theorem:
diagonal^2 = a^2 + a^2 = 2a^2 Taking the square root gives:
diagonal = a * √2 Thus the side length can be obtained by dividing the diagonal by √2:
a = diagonal / √2

Step-by-Step Solution:
Step 1: Let the side length of the square be a. Step 2: Use the relation diagonal = a√2 for a square. Step 3: The given diagonal length is 12 cm. Step 4: Set a√2 = 12. Step 5: Solve for a: a = 12 / √2. Step 6: Rationalise if desired: 12 / √2 = (12 * √2) / 2 = 6√2. Step 7: Therefore, the side length of the square is 6√2 cm.

Verification / Alternative check:
We can verify by squaring the side and checking the diagonal. If a = 6√2, then a^2 = 36 * 2 = 72. The sum of the squares of the two equal sides is 72 + 72 = 144, so the diagonal is sqrt(144) = 12 cm, which matches the given diagonal. This confirms the correctness of the side length.

Why Other Options Are Wrong:

• 12√2: This would make the diagonal equal to (12√2) * √2 = 24, not 12.
• 6: If the side were 6, then the diagonal would be 6√2, which is less than 12.
• 9 and 8: These values do not produce a diagonal of 12 cm when used in the diagonal formula for a square.

Common Pitfalls:
Students sometimes forget the √2 factor or misapply the Pythagorean theorem. Another common error is to assume the diagonal equals the sum of sides instead of applying the squared relationship. Remember the simple formula diagonal = side * √2 for a square to solve such problems quickly in exams.

Final Answer:
Thus, each side of the square measures 6√2 cm.