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

Introduction / Context:In a 30°–60°–90° right triangle, side lengths have a fixed ratio relative to the hypotenuse. Recognizing this pattern avoids trigonometry and speeds computation.

Given Data / Assumptions:

• Hypotenuse c = 9√3 cm.
• We need the side opposite the 30° angle.

Concept / Approach:In a 30°–60°–90° triangle: (short leg):(long leg):(hypotenuse) = 1 : √3 : 2. The side opposite 30° is the short leg, equal to hypotenuse/2.

Step-by-Step Solution:

Verification / Alternative check:If short leg = (9√3)/2, then long leg = short leg * √3 = (9√3/2)*√3 = (9*3)/2 = 27/2, and hypotenuse = 2 * short leg = 9√3, consistent with the given value.

Why Other Options Are Wrong:

• 9 cm, 6 cm: Treat √3 incorrectly or ignore the factor 1/2.
• 3√3 cm: This would be hypotenuse/3, not /2.

Common Pitfalls:Confusing which leg is opposite 30°; it is always the shorter leg.

Final Answer:(9√3)/2 cm