In right triangle XYZ, angle Y is 90°. Given that ∠Z = 60° and the hypotenuse ZX = 3√3 cm, find the exact length of the side YZ (opposite angle X) in centimetres.

Introduction / Context:
This problem uses properties of a right angled triangle with angles 30°, 60°, and 90°. The triangle XYZ is right angled at Y, and you are given one acute angle and the hypotenuse. The task is to find the length of one of the legs. Such questions rely on well known side ratios in 30°–60°–90° triangles and appear frequently in trigonometry and geometry sections of aptitude tests.

Given Data / Assumptions:

• Triangle XYZ is right angled at Y, so ∠Y = 90°.
• The angle at Z is ∠Z = 60°.
• Therefore the remaining angle at X is ∠X = 30° (since the angles of a triangle sum to 180°).
• The hypotenuse ZX has length 3√3 cm.
• We need to find the length of side YZ, which is opposite angle X.

Concept / Approach:
In any right triangle with angles 30°, 60°, and 90°, the sides have a fixed ratio. If the side opposite 30° is k, then the side opposite 60° is k√3, and the hypotenuse opposite 90° is 2k. Here the hypotenuse is known, so we can find k by dividing the hypotenuse by 2. Once k is known, we can identify YZ as the side opposite 30°, which has length k. Alternatively, we can use trigonometric definitions, for example sin 30° = opposite / hypotenuse, to compute the required side directly.

Step-by-Step Solution:
1) Since ∠Y = 90° and ∠Z = 60°, the remaining angle is ∠X = 180° − 90° − 60° = 30°. 2) In the 30°–60°–90° triangle, let the side opposite 30° be k, the side opposite 60° be k√3, and the hypotenuse be 2k. 3) Here the hypotenuse ZX corresponds to 2k and is given as 3√3 cm. 4) Set 2k = 3√3 and solve for k: k = (3√3) / 2. 5) The side YZ is opposite angle X, which is 30°, so YZ = k. 6) Therefore YZ = 3√3 / 2 cm.

Verification / Alternative check:
Use trigonometry as an alternative method. Since ∠X = 30°, we can write sin 30° = (opposite side to X) / (hypotenuse) = YZ / ZX. We know sin 30° = 1/2 and ZX = 3√3. So 1/2 = YZ / (3√3). Rearranging gives YZ = (1/2) * 3√3 = 3√3 / 2, which matches the result obtained from the 30°–60°–90° triangle side ratio. This confirms that the length of YZ is correctly computed.

Why Other Options Are Wrong:
Option b (3√3) would equal the hypotenuse, not the shorter leg opposite 30°. Option c (6) and option d (9) are too large, exceeding the hypotenuse or not consistent with the fixed side ratio. Option e (3/2) ignores the √3 factor and does not match the trigonometric relation sin 30° = 1/2. Only option a corresponds to the correct side length derived from both geometric and trigonometric reasoning.

Common Pitfalls:
One common error is mixing up which side is opposite which angle, especially between 30° and 60°. Another mistake is mis remembering the standard ratio, for example assuming the hypotenuse is k√3 instead of 2k. Students may also try to apply Pythagoras directly without using the known angle information, which works but is longer. Remembering the 1 : √3 : 2 side ratio for 30°–60°–90° triangles makes such questions quick and reliable.

Final Answer:
The side YZ in the right triangle XYZ has length 3√3/2 centimetres.