In right triangle DEF, the right angle is at E (∠E = 90°) and the measure of angle D is 30°. If the side opposite angle D is EF and EF = 6√3 cm, what is the length (in cm) of side DE?

Introduction:
This problem involves a standard 30°–60°–90° right triangle. Such triangles have fixed side length ratios, so knowing one side lets you compute the others quickly. Recognizing which side corresponds to which angle is the key step.

Given Data / Assumptions:

• Triangle DEF is right angled at E, so ∠E = 90°.
• ∠D = 30°, therefore ∠F = 60°.
• Side EF = 6√3 cm and EF is opposite angle D (30°).
• We must find DE, one of the other legs of the triangle.

Concept / Approach:
In a 30°–60°–90° triangle, if the side opposite 30° is a, then:

• Side opposite 60° is a√3.
• Hypotenuse (opposite 90°) is 2a.

By identifying EF as the side opposite 30°, we can set EF = a and use these ratios to find DE, which is the side opposite 60° in this configuration.

Step-by-Step Solution:
Angle D = 30°, so side opposite D is EF. Given EF = 6√3 cm, and in a 30°–60°–90° triangle that side equals a. So let a = 6√3. Side opposite 60° (which is DE) has length a√3. Thus DE = a√3 = 6√3 * √3. Compute √3 * √3 = 3, so DE = 6 * 3 = 18 cm.

Verification / Alternative check:
We can also find the hypotenuse DF = 2a = 2 * 6√3 = 12√3 cm. Check Pythagoras: DE² + EF² = 18² + (6√3)² = 324 + 108 = 432. Hypotenuse squared: (12√3)² = 144 * 3 = 432. Both match, so the triangle is consistent.

Why Other Options Are Wrong:
Options 12 and 12√3 do not align with the fixed ratios of sides in a 30°–60°–90° triangle given EF = 6√3. The option 18√3 is much too large; squaring it would violate the Pythagorean equality. Only 18 cm satisfies the standard triangle ratios and Pythagorean theorem simultaneously.

Common Pitfalls:
Students sometimes swap which side is opposite 30° and 60°, or incorrectly think the hypotenuse is a√3 instead of 2a. Keeping a clear picture of the angle side relationships prevents these mistakes.

Final Answer:
The length of DE is 18 cm.