I parked diagonally across the parking lot from the office door. My car is 5 feet vertically from the door and 10 feet horizontally from the door. To walk the shortest distance, should I walk along the sidewalk (red dashed line) or cut across the parking lot (yellow dashed line)? In other words, is the shortest distance between two points a straight line? Prove it by providing mathematical evidence. Your final answer should include a picture of your work and a statement, defended by mathematical evidence, declaring whether or not the shortest distance between two points is a straight line.

Question

I parked diagonally across the parking lot from the office door. My car is 5 feet vertically from the door and 10 feet horizontally from the door. To walk the shortest distance, should I walk along the sidewalk (red dashed line) or cut across the parking lot (yellow dashed line)? In other words, is the shortest distance between two points a straight line? Prove it by providing mathematical evidence. Your final answer should include a picture of your work and a statement, defended by mathematical evidence, declaring whether or not the shortest distance between two points is a straight line.

Answer 1

To determine the shortest distance between two points, we need to compare the lengths of the two different routes: walking along the sidewalk (red dashed line) and cutting across the parking lot (yellow dashed line).

Let's consider the two routes:
1) Walking along the sidewalk: The distance traveled can be calculated using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the vertical distance is 5 feet and the horizontal distance is 10 feet. Thus, the distance traveled along the sidewalk is √(5^2 + 10^2) = √(25 + 100) = √125 ≈ 11.18 feet.

2) Cutting across the parking lot: Since we are looking for the shortest distance, we assume a direct straight line from the car to the office door. In this case, the length of the direct line connecting the car to the office door can be calculated using the Pythagorean theorem as well.
Since the vertical distance is 5 feet and the horizontal distance is 10 feet, the length of the direct straight line across the parking lot is √(5^2 + 10^2) = √(25 + 100) = √125 ≈ 11.18 feet, which is the same as the distance along the sidewalk.

Based on the mathematical evidence, the distance along the sidewalk (red dashed line) and the direct straight line across the parking lot (yellow dashed line) are equal, both approximately 11.18 feet.

Therefore, in this specific scenario, there is no difference in taking either the sidewalk or cutting across the parking lot, as both routes cover the same distance. Therefore, the shortest distance between two points is indeed a straight line.

Attached is a diagram illustrating the scenario and the calculation.

[image: diagram attached]