In a right angled triangle, one acute angle measures 30 degrees and the side opposite this 30 degree angle has length 9 cm. What is the length of the side opposite the 60 degree angle in the same triangle, expressed in centimetres?

Introduction / Context:
This question is based on basic trigonometry and special right triangles. In particular, it uses the standard properties of a 30 60 90 right angled triangle, which is very common in aptitude and school level geometry. Understanding these fixed ratios helps you answer such questions quickly without using a calculator.

Given Data / Assumptions:
- The triangle is right angled and has angles 30 degrees, 60 degrees and 90 degrees.
- The side opposite the 30 degree angle is given as 9 cm.
- We are required to find the length of the side opposite the 60 degree angle.
- The triangle is assumed to be a standard 30 60 90 right triangle in Euclidean plane geometry.

Concept / Approach:
In a 30 60 90 right triangle, the sides are in a fixed ratio relative to the hypotenuse. If the hypotenuse has length 2k, then the side opposite 30 degrees has length k and the side opposite 60 degrees has length k√3. Another way to say this is that the side opposite 30 degrees is half of the hypotenuse, and the side opposite 60 degrees is √3 times the shorter leg. We will first find the hypotenuse using this ratio and then use it to get the side opposite 60 degrees.

Step-by-Step Solution:
Step 1: Let the hypotenuse be 2k and the side opposite 30 degrees be k in the standard 30 60 90 triangle. Step 2: According to the question, the side opposite 30 degrees is 9 cm. So k = 9. Step 3: The hypotenuse is therefore 2k = 2 * 9 = 18 cm. Step 4: The side opposite the 60 degree angle in such a triangle is k√3. Step 5: Substitute k = 9 to get the required side as 9√3 cm.

Verification / Alternative check:
You can also verify using trigonometric ratios. If the hypotenuse is 18 cm and the side opposite 30 degrees is 9 cm, then sin 30 degrees = 9 / 18 = 1 / 2, which is correct. For the 60 degree angle, sin 60 degrees should be (side opposite 60 degrees) / 18. Since sin 60 degrees is √3 / 2, we have side opposite 60 degrees = 18 * (√3 / 2) = 9√3 cm, which matches our result.

Why Other Options Are Wrong:
3√3 cm is too small because it would correspond to a different scale factor and would not maintain the 30 60 90 ratio with a 9 cm side opposite 30 degrees.
3/2 cm and 9/2 cm are both simple fractions of 9 but do not satisfy the fixed trigonometric ratio for a 30 60 90 triangle.
6√3 cm also does not fit the standard side length ratios when the side opposite 30 degrees is 9 cm.

Common Pitfalls:
A common mistake is to assume that the side opposite 60 degrees is double the side opposite 30 degrees or to confuse which side is the hypotenuse. Another frequent error is to forget the exact ratio of sides for a 30 60 90 triangle and try to guess from the options. Always remember the precise ratio of sides: k, k√3 and 2k for angles 30 degrees, 60 degrees and 90 degrees respectively.

Final Answer:
The length of the side opposite the 60 degree angle is 9√3 cm.