In right angled triangle DEF, the right angle is at E. If angle D is 30 degrees and side EF = 6√3 cm, what is the length of side DE in centimetres?

Introduction / Context:
This question is about a special right angled triangle with angles 30 degrees, 60 degrees, and 90 degrees. Such a triangle has well known side length ratios. We are given one side of the triangle and a specific angle, and we must find another side using these ratios. This is a standard trigonometry and geometry concept often tested in aptitude exams.

Given Data / Assumptions:

• Triangle DEF is right angled at E, so angle E = 90 degrees.
• Angle D = 30 degrees, so angle F = 60 degrees.
• Side EF = 6√3 cm.
• We need to find DE in cm.
• Triangle is therefore a 30-60-90 triangle.

Concept / Approach:
In a 30-60-90 right triangle, the sides opposite these angles follow the ratio 1 : √3 : 2. The smallest side, opposite 30 degrees, is k. The side opposite 60 degrees is k√3, and the hypotenuse opposite 90 degrees is 2k. Here, angle D is 30 degrees, and the side opposite angle D is EF. Thus EF corresponds to k. Using EF, we can determine k and then find DE, which is opposite 60 degrees and equals k√3.

Step-by-Step Solution:
Angle D = 30 degrees, angle E = 90 degrees, angle F = 60 degrees.In triangle DEF, side opposite angle D (30 degrees) is EF.Given EF = 6√3 cm.In a 30-60-90 triangle, let the side opposite 30 degrees be k, opposite 60 degrees be k√3, and hypotenuse be 2k.So EF corresponds to k and k = 6√3.Side opposite 60 degrees is DE = k√3 = 6√3 * √3 = 6 * 3 = 18 cm.Therefore DE = 18 cm.

Verification / Alternative check:
Having k = 6√3, hypotenuse DF would be 2k = 12√3. We can verify with the Pythagoras theorem. EF^2 = (6√3)^2 = 36 * 3 = 108. DE^2 = 18^2 = 324. Sum = 108 + 324 = 432. Hypotenuse squared DF^2 should be (12√3)^2 = 144 * 3 = 432, which matches. This confirms that DE = 18 cm fits the 30-60-90 triangle perfectly.

Why Other Options Are Wrong:
12√3 cm and 18√3 cm treat DE as hypotenuse or misassign the roles of sides in the 30-60-90 ratio. A value of 12 cm comes from confusing which side is opposite which angle. 9 cm is too short and does not satisfy Pythagoras when combined with EF = 6√3 and a consistent hypotenuse. Only 18 cm leads to a valid 30-60-90 triangle relation and a correct Pythagoras check.

Common Pitfalls:
Students often mix up which side corresponds to which angle in a 30-60-90 triangle. Remember that the side opposite 30 degrees is the smallest, opposite 60 degrees is medium, and opposite 90 degrees is the largest. Another error is assigning EF to the wrong part of the ratio. Carefully identifying the angle opposite each given side avoids these issues.

Final Answer:
18 cm