A man walks 9 km towards the east and then 12 km towards the south. Taking his starting point as reference, how far is he now from the starting point?

Introduction / Context:
This is a standard distance calculation question involving perpendicular movements. The man walks first east and then south, creating a right angled path. We need to calculate the straight line distance from his starting point to his final position. This is a direct application of the Pythagorean theorem to a right angled triangle with known legs.

Given Data / Assumptions:

• The man walks 9 km towards the east.
• He then walks 12 km towards the south.
• We assume a flat plane with east west and north south directions perpendicular to each other.
• We are asked for the straight line distance from the starting point to the final point, not the total path length walked.

Concept / Approach:
The path forms a right angled triangle whose legs are the eastward and southward distances. The hypotenuse of this triangle represents the crow flight distance between the starting and final points. The Pythagorean theorem states that, for a right angled triangle with legs a and b and hypotenuse c, we have c^2 = a^2 + b^2. Substituting a = 9 km and b = 12 km gives a straightforward calculation.

Step-by-Step Solution:
Step 1: Take the starting point as the origin (0, 0). After walking 9 km east, the man is at (9, 0).Step 2: After walking 12 km south, he is at (9, −12).Step 3: The displacement from the origin to (9, −12) has horizontal component 9 km and vertical component 12 km.Step 4: Apply the Pythagorean theorem: c^2 = 9^2 + 12^2 = 81 + 144 = 225.Step 5: Take the square root: c = √225 = 15 km.Step 6: Therefore, his straight line distance from the starting point is 15 km.

Verification / Alternative check:
We can notice that (9, 12, 15) is a scaled version of the classic (3, 4, 5) Pythagorean triple, multiplied by 3. In such triples, if the legs are 3 units and 4 units, the hypotenuse is 5 units. Here, the legs are 9 = 3 × 3 and 12 = 3 × 4, so the hypotenuse must be 3 × 5 = 15. This quick check confirms the more formal square and square root calculation and ensures that no arithmetic mistake has been made.

Why Other Options Are Wrong:
The option 8 km is smaller than both legs of the triangle and cannot be the hypotenuse. The option 6 km is even smaller and clearly impossible. The value 7.5 km might come from averaging the legs or mistakenly dividing something by 2, which is not correct here. The option 9 km corresponds only to the eastward leg and ignores the additional southward displacement. Only 15 km correctly reflects the combined effect of both perpendicular movements.

Common Pitfalls:
Some students mistakenly add the distances 9 and 12 and choose 21 km, confusing total path length with straight line distance. Others forget to square the numbers and incorrectly add them directly before taking a square root. Another error is to treat the triangle as isosceles or to guess based on rough proportions. Remembering the Pythagorean relationship and the common triples like 3, 4, 5 and their multiples is very helpful in such problems.

Final Answer:
The man is now at a straight line distance of 15 km from his starting point.