In triangle PQR, angle Q is 90 degrees (right angled at Q) and angle R is 60 degrees. If side RQ has length 4√3 centimetres, what is the length of the hypotenuse PR in centimetres?

Introduction / Context:
This problem uses the 30 60 90 special right triangle once again. When a right angled triangle has one 60 degree angle and one 30 degree angle, its side lengths follow a fixed ratio. Being comfortable with this ratio allows you to quickly find missing sides without using full trigonometric calculations every time.

Given Data / Assumptions:
- Triangle PQR is right angled at Q, so angle Q is 90 degrees.
- Angle R is 60 degrees, so angle P is 30 degrees since the sum of angles in a triangle is 180 degrees.
- Side RQ has length 4√3 cm.
- We need to find PR, which is the hypotenuse of the triangle.

Concept / Approach:
In a 30 60 90 triangle, if we denote the side opposite 30 degrees as k, then the side opposite 60 degrees is k√3, and the hypotenuse opposite 90 degrees is 2k. In triangle PQR, side QR is opposite angle P, which is 30 degrees. Therefore QR corresponds to k. Once we identify k from the given length, we can directly compute the hypotenuse PR as 2k.

Step-by-Step Solution:
Step 1: Recognise that angle P is 30 degrees because 30 + 60 + 90 = 180 degrees. Step 2: Identify side QR as the side opposite angle P, which is 30 degrees, so QR corresponds to k. Step 3: Given QR = 4√3 cm, we set k = 4√3. Step 4: The hypotenuse PR in a 30 60 90 triangle is 2k. Step 5: Substitute k = 4√3 to get PR = 2 * 4√3 = 8√3 cm.

Verification / Alternative check:
We can cross check using a trigonometric ratio. For angle R = 60 degrees, cos 60 degrees = adjacent / hypotenuse = RQ / PR. Since cos 60 degrees is 1 / 2, we get 1 / 2 = (4√3) / PR. Therefore PR = 8√3 cm, consistent with the special triangle ratio approach.

Why Other Options Are Wrong:
8 cm is too small, because if PR were 8 cm, then cos 60 degrees would be (4√3) / 8 = √3 / 2, not 1 / 2.
4 cm and 8 / √3 cm are both smaller than the given leg 4√3 cm, which is impossible for a hypotenuse.
12 cm does not satisfy the 30 60 90 side ratio when QR is 4√3 cm.

Common Pitfalls:
Learners sometimes incorrectly assign the role of k or k√3 to the wrong side by mixing up which side is opposite each angle. Another mistake is to try to use Pythagoras without identifying all sides, which can lead to extra steps and errors. Always mark the angles clearly and then map each side to the correct ratio in the 30 60 90 triangle pattern.

Final Answer:
The hypotenuse PR has length 8√3 cm.