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

Introduction:
This trigonometry based geometry question involves a right angled triangle with one acute angle of 30 degrees. It uses the special properties of 30°–60°–90° triangles, where side lengths maintain a fixed ratio. You are given one side and must find another using these relationships.

Given Data / Assumptions:

• Triangle ΔDEF is right angled at E, so ∠E = 90°.
• ∠D = 30°; hence ∠F = 60° (since the sum of angles in a triangle is 180°).
• Side EF = 6√3 cm.
• We need to find the length of side DE in centimetres.

Concept / Approach:
In a right triangle with angles 30°, 60°, and 90°, the side opposite 30° is the shortest, the side opposite 60° is √3 times the shortest side, and the hypotenuse is twice the shortest side. The ratio of sides opposite 30°, 60°, and 90° is: 1 : √3 : 2. We first identify which sides correspond to these angles, then use the ratio to compute DE.

Step-by-Step Solution:
Step 1: ∠D = 30°, ∠E = 90°, and ∠F = 60°. Step 2: Side EF is opposite ∠D (which is 30°), so EF is the side opposite 30°. Step 3: In the 1 : √3 : 2 ratio, let the side opposite 30° be x. Then: side opposite 60° = x√3 and hypotenuse = 2x. Step 4: Here, EF = x = 6√3? Actually, from the ratio, EF should equal x. But EF is given as 6√3, so x = 6√3. Step 5: Side DE is opposite ∠F (60°), so DE = x√3. Step 6: Substitute x = 6√3: DE = 6√3 * √3 = 6 * 3 = 18 cm.

Verification / Alternative check:
We can compute the hypotenuse DF = 2x = 2 * 6√3 = 12√3. Then check the Pythagorean theorem: DE^2 + EF^2 = 18^2 + (6√3)^2 = 324 + 36 * 3 = 324 + 108 = 432. Hypotenuse DF squared is (12√3)^2 = 144 * 3 = 432, so the values satisfy the Pythagorean theorem and are consistent.

Why Other Options Are Wrong:

• 12, 9: These are too small and do not satisfy the 1 : √3 : 2 ratio for the given EF.
• 12√3 and 18√3: These are larger than required and would not maintain the correct trigonometric relationships with EF = 6√3.

Common Pitfalls:
A frequent mistake is mixing up which side is opposite which angle. Remember that the side opposite 30° is the smallest, and the side opposite 60° is √3 times that side. Another error is miscomputing √3 * √3 as 3, so care must be taken when simplifying surds.

Final Answer:
The length of DE is 18 cm.