The diagonal of a square is 10 cm long. What is the length (in cm) of each side of the square?

Introduction / Context:
This is a straightforward geometry question involving the relationship between the side and diagonal of a square. It tests your ability to use Pythagoras theorem in a simple but important special case of a right angled isosceles triangle formed by the sides of a square and its diagonal.

Given Data / Assumptions:

• The figure is a square.
• The diagonal of the square is given as 10 cm.
• All sides of a square are equal, and each interior angle is 90 degrees.
• We must find the side length of the square.

Concept / Approach:
If the side of the square is s, then the diagonal is the hypotenuse of a right angled triangle whose legs are the two sides of the square. By Pythagoras theorem: Diagonal^2 = s^2 + s^2 = 2s^2 So: s^2 = (Diagonal^2) / 2 and s = Diagonal / √2 This formula is specific to a square and is a useful shortcut to remember.

Step-by-Step Solution:
Step 1: Let s be the side of the square (in cm). Step 2: The diagonal d is given as 10 cm. Step 3: Apply Pythagoras theorem to the right triangle formed by two sides and the diagonal: d^2 = s^2 + s^2 = 2s^2. Step 4: Substitute d = 10, so 10^2 = 2s^2, which gives 100 = 2s^2. Step 5: Divide by 2 to get s^2 = 50. Step 6: Therefore, s = √50 = √(25 * 2) = 5√2 cm.

Verification / Alternative check:
We can quickly verify by squaring the side: s = 5√2 gives s^2 = 25 * 2 = 50. Then the diagonal should be √(s^2 + s^2) = √(50 + 50) = √100 = 10 cm, which matches the given value.

Why Other Options Are Wrong:
5 cm: This would give a diagonal of √(5^2 + 5^2) = √50, which is less than 10 cm.
10√2 cm: This is larger than required and would correspond to a diagonal of √2 times the actual one.
5/√2 cm: This is smaller than 5 cm and does not produce a diagonal of 10 cm.
10 cm: This would incorrectly assume that side equals diagonal, which is never true in a square.

Common Pitfalls:
Some students reverse the roles and multiply the diagonal by √2 instead of dividing by √2. Others forget to apply Pythagoras theorem and try to use perimeter or area formulas. Remember that the diagonal of a square is always side times √2, so the side is diagonal divided by √2.

Final Answer:
The side length of the square is 5√2 cm.