P, Q and R are three towns. The distance between P and Q is 60 km, whereas the distance between P and R is 80 km. Q is in the West of P and R is in the South of P. What is the distance between Q and R?

The correct option is 100 km.

Let's break down the problem step-by-step using directional relationships and geometry.

Step 1: Understand the positions of the towns
We are given the relative directions of towns Q and R with respect to town P:
- Town P can be considered as the reference starting point (origin).
- Town Q is located to the West of Town P. This means the line segment PQ runs horizontally to the left from P.
- Town R is located to the South of Town P. This means the line segment PR runs vertically downwards from P.

Step 2: Identify the geometric shape
Since West and South are perpendicular directions, the angle between the line segment PQ (West direction) and the line segment PR (South direction) is exactly 90° (a right angle).
Therefore, the points P, Q, and R form a right-angled triangle, $\Delta QPR$, with the right angle at vertex P ($\angle QPR=90°$).

Step 3: Apply the Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, which is the distance between Q and R) is equal to the sum of the squares of the other two sides (PQ and PR).
The Pythagorean formula is:

${QR}^2={PQ}^2+{PR}^2$

Given values:
- Distance between P and Q (PQ) = 60 km
- Distance between P and R (PR) = 80 km

Substituting these values into the formula:

${QR}^2={60}^2+{80}^2$

${QR}^2=3600+6400$

${QR}^2=10000$

Taking the square root on both sides to find QR:

$QR=\sqrt{10000}$

$QR=100\text{km}$

Thus, the distance between town Q and town R is 100 km.