In right angled triangle PQR with angle R = 90 degrees, RS is drawn perpendicular to hypotenuse PQ. If PR = 3 cm and RQ = 4 cm, what is the length of RS in centimetres?

Introduction / Context:
This question involves a standard right triangle configuration and the altitude drawn from the right angle to the hypotenuse. Such problems frequently appear in aptitude tests to check if you remember special formulas and can apply Pythagoras theorem quickly. The relation between the legs, hypotenuse and the altitude from the right angle is very useful in many geometric situations.

Given Data / Assumptions:
• Triangle PQR is right angled at R, so ∠R = 90 degrees. • PR = 3 cm and RQ = 4 cm are the legs of the right triangle. • RS is drawn perpendicular to the hypotenuse PQ, so RS is the altitude from the right angle. • We need to find the length of RS.

Concept / Approach:
For a right triangle with legs a and b and hypotenuse c, the altitude h from the right angle to the hypotenuse satisfies the relation h = (a * b) / c. This can be derived from similarity of triangles or from area formulas. First we find the hypotenuse PQ using the Pythagoras theorem: c^2 = a^2 + b^2. Then we use the altitude formula to compute RS directly. This approach is efficient and avoids more complicated constructions.

Step-by-Step Solution:
1. Identify the legs: PR = 3 cm and RQ = 4 cm. 2. Compute the hypotenuse PQ using Pythagoras theorem: PQ^2 = PR^2 + RQ^2. 3. So PQ^2 = 3^2 + 4^2 = 9 + 16 = 25. 4. Therefore, PQ = sqrt(25) = 5 cm. 5. Let RS = h, the altitude from R to PQ. 6. Use the formula h = (a * b) / c, where a = PR, b = RQ and c = PQ. 7. Substitute the values: h = (3 * 4) / 5. 8. Compute h = 12 / 5 cm. 9. So RS = 12 / 5 cm, which is 2.4 cm in decimal form.

Verification / Alternative check:
Another way to verify is to compare areas. The area of triangle PQR is (1 / 2) * PR * RQ = (1 / 2) * 3 * 4 = 6 square centimetres. The same area can also be expressed as (1 / 2) * PQ * RS = (1 / 2) * 5 * RS. Equating these gives 6 = (1 / 2) * 5 * RS, which leads to RS = 12 / 5 cm. This matches the earlier result and confirms the correctness of our method.

Why Other Options Are Wrong:
• 36/5 cm: This is much larger than the hypotenuse and cannot be the altitude of such a small triangle. • 5 cm: This equals the hypotenuse length, which is impossible for a perpendicular segment inside the triangle. • 2.5 cm: Close to 2.4 cm but not exact; it comes from rough rounding rather than correct calculation. • 3 cm: This matches one leg but has no geometric justification as the altitude to the hypotenuse.

Common Pitfalls:
Some students mistakenly treat the altitude as equal to one of the legs or use the wrong formula h^2 = a * b without checking the context. Another common error is to forget to find the hypotenuse first and plug incorrect values into the altitude formula. Always remember to apply Pythagoras theorem carefully and keep track of which side is which.

Final Answer:
The length of RS is 12/5 cm.