In right angled triangle XYZ, angle Y is 90° and angle Z is 30°. If the hypotenuse ZX is 9 cm, what is the length (in cm) of side YZ?

Introduction / Context:
This trigonometry question uses special angle values in a right angled triangle. When one acute angle is 30° and the other is 60°, the triangle is a 30 60 90 special triangle with fixed ratios between its sides. Knowing these ratios allows you to find any side given one of them, especially the hypotenuse.

Given Data / Assumptions:

• Triangle XYZ is right angled at Y, so angle Y = 90°.
• Angle Z = 30°, so angle X = 60° (because the sum of angles in a triangle is 180°).
• The hypotenuse is ZX, and ZX = 9 cm.
• We must find the length of side YZ.
• YZ is a leg of the triangle adjacent to angle Z and opposite angle X.

Concept / Approach:
In a 30 60 90 triangle, the side opposite 30° is half the hypotenuse, the side opposite 60° is (√3 / 2) times the hypotenuse, and the hypotenuse is the longest side. Here, the hypotenuse is ZX and angle Z is 30°, so:

• Side opposite 30° (which is YX) = (1 / 2) * hypotenuse.
• Side opposite 60° (which is YZ) = (√3 / 2) * hypotenuse.

We need the side opposite 60°, so we multiply the hypotenuse by √3 / 2.

Step-by-Step Solution:
Step 1: Recognize that the triangle is a 30 60 90 special triangle because one angle is 30° and the right angle is 90°. Step 2: Identify side roles: ZX is the hypotenuse opposite the right angle at Y. Step 3: Side YX is opposite angle Z = 30° and side YZ is opposite angle X = 60°. Step 4: In a 30 60 90 triangle, side opposite 60° = (√3 / 2) * hypotenuse. Step 5: Substitute hypotenuse ZX = 9 cm: YZ = (√3 / 2) * 9 = 9√3 / 2 cm. Step 6: So the length of YZ is 9√3 / 2 cm.

Verification / Alternative check:
We can also find the other leg YX. Side opposite 30° is half the hypotenuse, so YX = (1 / 2) * 9 = 4.5 cm. Now check Pythagoras theorem: (YX)^2 + (YZ)^2 should equal hypotenuse^2. Compute YX^2 = 4.5^2 = 20.25 and YZ^2 = (9√3 / 2)^2 = 81 * 3 / 4 = 243 / 4 = 60.75. Sum: 20.25 + 60.75 = 81, which equals 9^2 = 81, confirming the correctness of the side lengths.

Why Other Options Are Wrong:
6√3 and 3√3 correspond to incorrect multiples of the hypotenuse and do not respect the special 30 60 90 triangle ratios. 18 cm would be twice the hypotenuse and cannot be a leg in a right triangle. 4.5 cm is the side opposite 30°, not the side opposite 60°, so it is the other leg, not YZ.

Common Pitfalls:
A frequent mistake is to mix up which side corresponds to which angle in the 30 60 90 triangle. Some students incorrectly take the side opposite 30° as (√3 / 2) times the hypotenuse and the side opposite 60° as half, which is backwards. Always remember: the smaller angle (30°) faces the smaller leg (half the hypotenuse), and the larger acute angle (60°) faces the longer leg (√3 / 2 times the hypotenuse).

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