In right triangle DEF, right angled at E, angle D is 30°. If side EF = 2√3 cm, what is the length of side DE (in cm)?

Introduction / Context: Right triangles with one angle equal to 30° are common in aptitude tests because they lead to standard side ratios. A 30°–60°–90° triangle has a fixed relationship between its sides, and knowing this pattern allows you to solve problems quickly without trigonometric tables. This question uses that standard triangle to relate a given side to another unknown side.

Given Data / Assumptions:

• Triangle DEF is a right triangle with right angle at E.
• Angle D is 30°, so angle F is 60°.
• Side EF, which is opposite angle D, has length 2√3 cm.
• We must find the length of side DE.

Concept / Approach: In a 30°–60°–90° triangle, the sides follow a fixed ratio. If the side opposite the 30° angle is a, then: hypotenuse = 2a side opposite 60° = a√3 Here, angle D is 30°, and the side opposite D is EF. Hence EF is the smallest side a. The hypotenuse is DF, and the remaining leg is DE. Since EF is opposite 30°, DE is opposite 60° and should equal a√3.

Step-by-Step Solution: Step 1: Identify EF as the side opposite the 30° angle at D. Step 2: Let a be the length of the side opposite 30°. So EF = a. Step 3: We are given EF = 2√3 cm, so a = 2√3. Step 4: In a 30°–60°–90° triangle, the side opposite 60° (here DE) is a√3. Step 5: Compute DE = a√3 = 2√3 * √3. Step 6: Simplify √3 * √3 = 3, so DE = 2 * 3 = 6. Step 7: Therefore, DE = 6 cm.

Verification / Alternative check: We can check using the Pythagorean theorem. With EF = 2√3 and DE = 6, compute DF, the hypotenuse. Then: DF^2 = DE^2 + EF^2 = 6^2 + (2√3)^2 = 36 + 4 * 3 = 36 + 12 = 48. So DF = sqrt(48) = 4√3. In a 30°–60°–90° triangle where the shortest side is a = 2√3, the hypotenuse should be 2a = 4√3, which matches the computed value. This confirms that DE = 6 is correct.

Why Other Options Are Wrong:

• 3 and 4: These do not satisfy the standard side ratios for a 30°–60°–90° triangle given EF = 2√3.
• 2: This is smaller than the side opposite 30°, which is not possible for the side opposite 60°.
• 2√3: This repeats the smallest side and ignores the correct scaling by √3 for the 60° side.

Common Pitfalls: A common mistake is to mislabel which angle is 30° and which side is opposite it, leading to incorrect assignment of the smallest side. Others may try to use the Pythagorean theorem directly but make algebraic errors. Memorising the 30°–60°–90° side ratio and mapping the given data correctly is the fastest and safest method.

Final Answer: Thus, the length of side DE is 6 cm.