In a right triangle, the length of the side opposite the right angle (the hypotenuse) is 9√3 cm. What is the length of the side opposite the angle that measures 30°?

Introduction / Context:
This question deals with a special right triangle that has angles 30°, 60°, and 90°. Such triangles have well known side ratios that greatly simplify calculations. Recognizing this pattern allows us to find the length of a specific side directly from the length of the hypotenuse, without using heavy trigonometry.

Given Data / Assumptions:

• The triangle is a right triangle with one angle of 30° and another of 60°.
• The side opposite the right angle, the hypotenuse, is 9√3 cm.
• We are asked for the side opposite the 30° angle.
• The triangle follows standard Euclidean geometry.
• We may use the standard ratios for a 30°–60°–90° triangle.

Concept / Approach:
In a 30°–60°–90° triangle, the side opposite 30° is the shortest side and has length x, the side opposite 60° has length x√3, and the hypotenuse has length 2x. Here we are given the hypotenuse and need to find x. Once we find x, that value directly gives the side opposite 30°. The problem reduces to solving 2x equal to the known hypotenuse length.

Step-by-Step Solution:
Let the side opposite 30° be x.In a 30°–60°–90° triangle, hypotenuse = 2x.Given hypotenuse = 9√3 cm, so 2x = 9√3.Solve for x: x = (9√3) / 2.Therefore, the side opposite the 30° angle is (9√3) / 2 cm.

Verification / Alternative check:
We can check by computing the other leg. The side opposite 60° should be x√3 = ((9√3) / 2) * √3 = (9 * 3) / 2 = 27 / 2 = 13.5 cm. Now, apply Pythagoras theorem: (side opposite 30°)^2 + (side opposite 60°)^2 should equal hypotenuse^2. Compute ((9√3) / 2)^2 + (13.5)^2 and verify it equals (9√3)^2. This check confirms that the side lengths are consistent with a 30°–60°–90° triangle.

Why Other Options Are Wrong:
Option 9 cm would make the hypotenuse 18 cm, which is inconsistent with 9√3 cm. Option 3√3 cm is too small and does not satisfy the 2x relation with the hypotenuse. Option 6 cm would produce a hypotenuse of 12 cm, again inconsistent. Option (3√3)/2 cm is smaller still and does not fit the triangle side ratio system for the given hypotenuse.

Common Pitfalls:
Some students confuse which side corresponds to which angle and may assign the value to the wrong side. Others misremember the ratio as 1 : 2 : √3 instead of 1 : √3 : 2. A few may try to use sine or cosine without recognizing the standard pattern, leading to extra steps and potential arithmetic errors.

Final Answer:
The length of the side opposite the 30° angle is (9√3)/2 cm.