Two vertical buildings have heights of 38 metres and 58 metres respectively. The straight-line distance between their tops is measured to be 52 metres. Using this information and applying the Pythagoras theorem, what is the horizontal distance (in metres) between the two buildings at ground level?

Introduction / Context:
This question tests the learner on the concept of right triangles and the Pythagoras theorem, which is a standard topic in geometry and also appears frequently in quantitative aptitude. The heights of two buildings and the straight-line distance between their tops together form a right triangle with the horizontal ground distance between the buildings as the base and the difference in their heights as the vertical side.

Given Data / Assumptions:

Height of the first building = 38 metres.
Height of the second building = 58 metres.
Straight-line distance between the tops of the buildings = 52 metres.
The buildings are vertical and stand on the same horizontal level ground.
We assume the line joining the tops of the buildings and the line joining their bases form a right triangle with the vertical difference in height.

Concept / Approach:
If two vertical structures stand on a flat horizontal surface, the line connecting their tops and the line connecting their bases form the hypotenuse and the base of a right triangle. The difference in heights forms the vertical side. The Pythagoras theorem states that in a right triangle, hypotenuse^2 = base^2 + height^2. Here, the hypotenuse is the distance between the tops, the vertical side is the difference in heights, and the horizontal distance between the buildings is the unknown base we need to determine.

Step-by-Step Solution:
Step 1: Compute the difference in the heights of the two buildings: 58 − 38 = 20 metres. Step 2: Consider the right triangle where the hypotenuse is 52 metres and one side (vertical) is 20 metres. Step 3: Let the horizontal distance between the buildings be d metres. Then, by Pythagoras theorem, d^2 + 20^2 = 52^2. Step 4: Calculate d^2: 52^2 = 2704 and 20^2 = 400, so d^2 = 2704 − 400 = 2304. Step 5: Take the square root of 2304: d = √2304 = 48 metres.

Verification / Alternative check:
We can verify by checking Pythagoras theorem again. If the base is 48 metres and the vertical side is 20 metres, then hypotenuse^2 = 48^2 + 20^2 = 2304 + 400 = 2704. The square root of 2704 is 52, which matches the given distance between the tops of the buildings. This confirms that our computed horizontal distance of 48 metres is correct.

Why Other Options Are Wrong:
A distance of 42 metres or 44 metres would result in hypotenuse values that are not equal to 52 metres when substituted into the Pythagoras relation. A distance of 46 metres also fails to satisfy the equation d^2 + 20^2 = 52^2. Only d = 48 metres fits the numerical requirement exactly and is consistent with the geometry of the problem.

Common Pitfalls:
A frequent mistake is to add the heights instead of taking their difference, which would incorrectly use 38 + 58 instead of 58 − 38. Another error is to assume that the 52 metres is the base instead of the hypotenuse, which reverses the roles of the sides. Learners also sometimes miscalculate squares of numbers or square roots. Careful use of the Pythagoras theorem and accurate arithmetic leads to the correct result.

Final Answer:
The horizontal distance between the two buildings is 48 metres.