In right angled triangle ABC, angle B is 90 degrees and angle A is 30 degrees. If the hypotenuse AC has length 8 centimetres, what is the length of side AB in centimetres?

Introduction / Context:
This problem again uses the special properties of a 30 60 90 right angled triangle. When one of the acute angles in a right triangle is 30 degrees, the ratios of the side lengths become fixed. Remembering these ratios makes it much faster to solve questions without repeatedly using trigonometric functions.

Given Data / Assumptions:
- Triangle ABC is right angled at B, so angle B is 90 degrees.
- Angle A is 30 degrees, so angle C must be 60 degrees.
- The hypotenuse is AC with length 8 cm.
- We need to find side AB, which is one of the legs of the triangle.

Concept / Approach:
In a 30 60 90 triangle, the side opposite 30 degrees is the smallest, the side opposite 60 degrees is medium and the side opposite 90 degrees is the hypotenuse. If the hypotenuse is 2k, then the side opposite 30 degrees is k and the side opposite 60 degrees is k√3. In this triangle, side BC is opposite angle A, which is 30 degrees, and side AB is opposite angle C, which is 60 degrees. Knowing the hypotenuse AC, we can first find k and then compute the required side AB.

Step-by-Step Solution:
Step 1: Let the hypotenuse AC be 2k in the standard 30 60 90 triangle. Step 2: Given AC = 8 cm, so 2k = 8, which gives k = 4. Step 3: The side opposite 60 degrees is k√3. Here angle C is 60 degrees, and its opposite side is AB. Step 4: Substitute k = 4 to get AB = 4√3 cm. Step 5: Hence the required side length AB is 4√3 cm.

Verification / Alternative check:
We can use a trigonometric ratio such as sine. Since angle A is 30 degrees, sin 30 degrees = opposite / hypotenuse = BC / AC. This gives BC = 8 * 1 / 2 = 4 cm. Then side AB, which is opposite the 60 degree angle at C, must satisfy Pythagoras theorem: AB^2 = AC^2 - BC^2 = 8^2 - 4^2 = 64 - 16 = 48, so AB = √48 = 4√3 cm. This is consistent with the special triangle ratio method.

Why Other Options Are Wrong:
2√3 cm and 2 / √3 cm are too small and do not satisfy Pythagoras theorem with hypotenuse 8 cm.
4 / √3 cm is also too small and would give a hypotenuse less than 8 cm when combined with the other leg.
8√3 cm is far too large to be a leg when the hypotenuse is only 8 cm.

Common Pitfalls:
A common mistake is to take the side opposite 30 degrees as k√3 instead of k, thereby reversing the ratios. Another error is to confuse which side is opposite which angle when the triangle is named. Drawing a quick sketch and marking each angle and its opposite side can help avoid confusion.

Final Answer:
The length of side AB is 4√3 cm.