In right triangle ΔXYZ, which is right angled at Y, if ∠Z = 60° and the hypotenuse ZX = 9√3 cm, then what is the length (in centimetres) of side YZ?

Introduction / Context:
This problem uses the properties of a 30 60 90 right triangle. In such a triangle, the lengths of the sides have fixed ratios, and recognising this pattern allows you to solve quickly without relying solely on the Pythagorean theorem or trigonometric functions every time.

Given Data / Assumptions:

• Triangle ΔXYZ is right angled at Y.
• Angle Z is 60 degrees, so angle X is 30 degrees.
• Hypotenuse ZX has length 9√3 cm.
• We need the length of side YZ, which is opposite angle X (30 degrees).

Concept / Approach:
In a 30 60 90 right triangle, the side opposite 30 degrees is the shortest side, denoted s. The side opposite 60 degrees is s√3, and the hypotenuse is 2s. With the hypotenuse given, we first determine s, then identify which side in the given triangle corresponds to s to find YZ. This ratio based method is efficient and avoids long calculations.

Step-by-Step Solution:
Because the triangle is right angled at Y and ∠Z = 60°, the remaining angle ∠X is 30°. Thus ΔXYZ is a 30 60 90 triangle. In such a triangle, if the side opposite 30° is s, then the hypotenuse is 2s. The hypotenuse ZX is given as 9√3 cm. Therefore 2s = 9√3, so s = (9√3)/2. The side opposite 30° is YZ, so YZ = s = 9√3/2 cm.

Verification / Alternative check:
We can check using the Pythagorean theorem. The side opposite 60° (XY) is s√3 = (9√3/2) * √3 = (9 * 3)/2 = 27/2. Then YZ^2 + XY^2 must equal ZX^2. Compute YZ^2 = (9√3/2)^2 = 81 * 3 / 4 = 243/4. Compute XY^2 = (27/2)^2 = 729/4. Adding gives 243/4 + 729/4 = 972/4 = 243. Now ZX^2 = (9√3)^2 = 81 * 3 = 243. The equality holds, confirming the correctness of YZ = 9√3/2.

Why Other Options Are Wrong:
Values such as 3√3 or 3√3/2 are smaller than the correct side and would give a hypotenuse different from 9√3 when checked. The value √3 is far too small, and 9 cm would correspond to a different triangle ratio. Only 9√3/2 satisfies the 30 60 90 triangle ratios and passes the Pythagorean verification.

Common Pitfalls:
Learners sometimes mix up which side is opposite which angle, especially when drawing the triangle quickly. Others reverse the ratio, treating the hypotenuse as s and the short leg as 2s, which leads to incorrect lengths. Drawing a clear diagram and labelling sides by angle can significantly reduce confusion.

Final Answer:
The length of side YZ is 9√3/2 cm.