In a right angled triangle, the side opposite the right angle (the hypotenuse) is 6 cm long. What is the length of the side opposite to a 30° angle in this triangle?

Introduction / Context:
This question uses a standard result from right triangle trigonometry, specifically the special 30°–60°–90° triangle. Many aptitude and entrance exams use this property to create fast questions where knowledge of fixed ratios saves time compared to full trigonometric calculations.

Given Data / Assumptions:
- The triangle is right angled, so one angle is 90°.
- The side opposite the right angle, which is the hypotenuse, has length 6 cm.
- One acute angle is 30°.
- We need the length of the side opposite the 30° angle.

Concept / Approach:
In any 30°–60°–90° right triangle the side lengths follow a fixed ratio:
- Side opposite 30° : side opposite 60° : hypotenuse = 1 : √3 : 2.
This means the side opposite 30° is exactly half of the hypotenuse. Therefore, once we know the hypotenuse, we can immediately find the side opposite 30° by dividing the hypotenuse by 2.

Step-by-Step Solution:
Step 1: Recognize the triangle as a 30°–60°–90° right triangle. Step 2: Use the standard ratio: side opposite 30° = hypotenuse / 2. Step 3: The hypotenuse is given as 6 cm, so the side opposite 30° is 6 / 2 = 3 cm. Step 4: Thus the required side length is 3 cm.

Verification / Alternative Check:
If the side opposite 30° is 3 cm, then using the ratio 1 : √3 : 2, the side opposite 60° should be 3√3 cm, and the hypotenuse should be 2 * 3 = 6 cm. This matches the given hypotenuse, confirming that 3 cm for the side opposite 30° is consistent with the known geometric ratio.

Why Other Options Are Wrong:
Option b, 3√3 cm, is the length of the side opposite the 60° angle, not the 30° angle.
Option c, 3/2 cm, is one quarter of the hypotenuse and does not correspond to any standard ratio here.
Option d, (3/2)√3 cm, also does not match the known 1 : √3 : 2 ratio when matched to a hypotenuse of 6 cm.

Common Pitfalls:
A frequent mistake is mixing up which leg corresponds to 30° and which to 60°. Another pitfall is attempting to use trigonometric functions without recognizing the special triangle pattern, which takes more time and increases the chance of calculator or algebra errors. Remembering the simple 1 : √3 : 2 ratio for 30°–60°–90° triangles gives very quick solutions.

Final Answer:
The length of the side opposite the 30° angle is 3 cm.