Mr. X starts from point A, travels 80 km towards the west, then takes a left turn, travels 50 km and reaches point B. What is the shortest straight line distance between points A and B?

Introduction / Context:
This problem converts a path with a turn into a right triangle and asks for the shortest distance between the starting point A and the final point B. Mr. X travels west and then south (after a left turn from west). The straight line joining A and B is the hypotenuse of a right angled triangle. Such questions link direction sense with basic Pythagoras theorem and distance calculation.

Given Data / Assumptions:
- Mr. X starts from point A. - He travels 80 km towards the west. - From there he takes a left turn. When facing west, a left turn points towards the south. - He then travels 50 km south and reaches point B. - The ground is flat and distances are measured along straight lines.

Concept / Approach:
The two legs of travel form perpendicular sides of a right triangle: one horizontal (west) and one vertical (south). The shortest straight line distance from A to B is the hypotenuse of this right triangle. If one side has length a and the other has length b, the hypotenuse is sqrt(a^2 + b^2). Here, a = 80 and b = 50. We compute c = sqrt(80^2 + 50^2) and then simplify the square root to match the options written in the form 10 * sqrt(n).

Step-by-Step Solution:
1. Represent A as (0, 0). 2. After travelling 80 km west, Mr. X reaches (-80, 0). 3. Facing west, a left turn points south, so he travels 50 km south to (-80, -50), which is point B. 4. The horizontal difference between A and B is 80 km and the vertical difference is 50 km. 5. The shortest straight line distance AB is given by distance = sqrt(80^2 + 50^2). 6. Compute 80^2 = 6400 and 50^2 = 2500, so distance^2 = 6400 + 2500 = 8900. 7. Therefore distance = sqrt(8900) = sqrt(89 * 100) = 10 * sqrt(89) km.

Verification / Alternative check:
We can cross check by comparing with the options. The options are of the form 10 * sqrt(n). We know that the hypotenuse must be greater than the longer side but less than the sum of both legs: 80 km < AB < 130 km. Evaluating the approximate values, 10 * sqrt(89) is about 10 * 9.43 ≈ 94.3 km, which fits within this range. The other options correspond to different sums of squares and would not equal 8900 when squared. Since only 10 * sqrt(89) km satisfies the exact calculation, it must be the correct choice.

Why Other Options Are Wrong:
- 10 * sqrt(39) km would give distance^2 = 3900, which is far less than the required 8900. - 10 * sqrt(98) km corresponds to distance^2 = 9800, not equal to 80^2 + 50^2. - 10 * sqrt(93) km corresponds to distance^2 = 9300, which again does not match 8900. - 50 km is merely one leg of the triangle and not the combined hypotenuse.

Common Pitfalls:
Common mistakes include adding the distances (80 + 50) instead of using the Pythagoras theorem, or squaring and adding incorrectly. Some test takers also misinterpret the left turn direction from west. Always confirm that the two legs are perpendicular and then apply distance = sqrt(a^2 + b^2). Checking approximate values of square roots can provide a quick sanity check against unreasonable answers.

Final Answer:
The shortest distance between points A and B is 10 * sqrt(89) km.