In a square, the length of the diagonal is 12 cm. Using properties of squares and right triangles, what is the length (in cm) of each side of the square?

Introduction:
This question uses the well known relationship between the side and the diagonal of a square. Drawing a diagonal splits the square into two congruent right triangles, and the Pythagorean theorem gives a direct connection between the side length and diagonal length.

Given Data / Assumptions:

• The figure is a square.
• The diagonal of the square is 12 cm.
• All sides of a square are equal, and each interior angle is 90°.
• We must find the side length of the square.

Concept / Approach:
In a square with side length s, the diagonal d forms the hypotenuse of a right triangle whose legs are both s. By the Pythagorean theorem, d² = s² + s² = 2s², so d = s√2. Solving for s gives s = d / √2. This formula can be used directly once the diagonal is known.

Step-by-Step Solution:
Let side length of the square be s. Diagonal d = 12 cm. Relationship: d = s√2. So 12 = s√2. Solve for s: s = 12 / √2. Rationalize if desired: 12 / √2 = (12√2) / 2 = 6√2. Therefore, each side of the square is 6√2 cm long.

Verification / Alternative check:
Check by reversing the relationship: if s = 6√2, then diagonal d should be s√2 = 6√2 * √2 = 6 * 2 = 12 cm, which matches the given diagonal length.

Why Other Options Are Wrong:
Values 6 cm or 9 cm would give diagonals of 6√2 cm or 9√2 cm, not 12 cm. The option 12√2 cm is far too large for the side, as its diagonal would then be 12√2 * √2 = 24 cm. Only 6√2 cm reproduces the given diagonal of 12 cm.

Common Pitfalls:
Students sometimes incorrectly assume the diagonal equals 2s instead of s√2, confusing squares with rectangles of special dimensions. Another mistake is forgetting to divide by √2 or rationalizing incorrectly. Always derive the relationship from the Pythagorean theorem if unsure.

Final Answer:
The length of each side of the square is 6√2 cm.