In a triangle, the side opposite an angle of 60° measures 6√3 cm. Assuming the triangle is a right angled triangle with one angle equal to 90°, what is the length of the side opposite the 90° angle?

Introduction / Context:
This question is based on the special properties of a 30°–60°–90° right triangle. We are told that one interior angle is 60° with a side opposite this angle equal to 6√3 cm, and another angle is 90°, revealing that the triangle is right angled. In such a triangle, the side lengths follow fixed ratios relative to one another. Using these known ratios, we can quickly determine the length of the side opposite the 90° angle, which is the hypotenuse.

Given Data / Assumptions:
- The triangle is right angled, with one angle equal to 90°. - Another angle is 60°, so the remaining angle must be 30°. - The side opposite the 60° angle is given as 6√3 cm. - We are asked to find the side opposite the 90° angle (the hypotenuse). - All sides are positive real lengths.

Concept / Approach:
In a 30°–60°–90° triangle, side lengths are in a fixed proportional pattern. If the side opposite the 30° angle is taken as k, then the side opposite the 60° angle is k√3 and the side opposite the 90° angle (the hypotenuse) is 2k. This ratio holds for all such triangles, regardless of their absolute size. In this problem, the side opposite 60° is given, so we can solve for k and then double it to find the hypotenuse opposite 90°.

Step-by-Step Solution:
Step 1: Recognise that with angles 30°, 60° and 90°, the triangle is a special right triangle. Step 2: Let the side opposite the 30° angle be k. Step 3: Then the side opposite the 60° angle is k√3, and the side opposite the 90° angle is 2k. Step 4: We are given that the side opposite the 60° angle is 6√3 cm. Thus k√3 = 6√3. Step 5: Cancel √3 from both sides to obtain k = 6. Step 6: The hypotenuse, which is the side opposite the 90° angle, is 2k. Step 7: Compute 2k = 2 * 6 = 12 cm. Step 8: Therefore, the length of the side opposite the 90° angle is 12 cm.

Verification / Alternative check:
Once we know k = 6, the side opposite 30° is 6 cm, the side opposite 60° is 6√3 cm, and the hypotenuse is 12 cm. We can verify the Pythagoras theorem: for the legs 6 and 6√3, the square of the hypotenuse should equal 6^2 + (6√3)^2 = 36 + 36 * 3 = 36 + 108 = 144. The square of 12 is also 144, so the triangle is indeed right angled and consistent with the given side length. This confirms that 12 cm is correct.

Why Other Options Are Wrong:
Option 12√3 cm would make the hypotenuse much larger and would not satisfy the 30°–60°–90° ratio with the given side of 6√3 cm. Option 6 cm corresponds to the side opposite the 30° angle, not the side opposite the 90° angle. Option 3√3 cm is too small and would not be consistent with the given side 6√3 cm in a right triangle.

Common Pitfalls:
A frequent mistake is mixing up which side in a 30°–60°–90° triangle is associated with k and which with k√3. Some learners also incorrectly assume that the side opposite 60° is the longest, but in a right triangle the hypotenuse, opposite 90°, is always the longest side. Remembering and applying the fixed ratio 1 : √3 : 2 for the sides opposite 30°, 60° and 90° respectively helps to avoid such confusion.

Final Answer:
The length of the side opposite the 90° angle is 12 cm.