Understanding Maheep's Walk and Shortest Distance

This problem involves calculating distances based on a series of movements and the overall shortest distance (displacement) between the start point (office) and the end point (home). Let's break down Maheep's path step by step.

Imagine the office is at the origin (0,0) on a coordinate plane. South is in the negative y direction, East is in the positive x direction, West is in the negative x direction, and North is in the positive y direction.

• Step 1: Maheep walks 120 m towards south from the office.
• Current position: (0, -120).
• Step 2: He turns left. From South, turning left means walking East. He walks a distance of A m.
• Current position: (0 + A, -120) = (A, -120).
• Step 3: He turns right. From East, turning right means walking South. He walks 80 m.
• Current position: (A, -120 - 80) = (A, -200).
• Step 4: He turns left. From South, turning left means walking East. He walks another distance of B m. He reaches his home.
• Home position: (A + B, -200).

So, the starting point (office) is (0, 0) and the ending point (home) is (A + B, -200). The shortest distance between his office and home is given as 290 m.

Calculating Shortest Distance Using Pythagorean Theorem

The shortest distance between two points in a coordinate plane is the straight line connecting them. This straight line forms the hypotenuse of a right-angled triangle, where the legs are the total displacement in the horizontal (East-West) and vertical (North-South) directions.

• Total horizontal displacement = Final x-coordinate - Initial x-coordinate = (A + B) - 0 = A + B.
• Total vertical displacement = Final y-coordinate - Initial y-coordinate = -200 - 0 = -200. The magnitude of vertical displacement is $|-200| = 200$.

According to the Pythagorean theorem, the square of the hypotenuse (shortest distance) is equal to the sum of the squares of the other two sides (horizontal and vertical displacements).

Let D be the shortest distance. So, \(D = 290\) m.

\( (\text{Total horizontal displacement})^2 + (\text{Total vertical displacement})^2 = (\text{Shortest distance})^2 \)

\( (A + B)^2 + (200)^2 = (290)^2 \)

Solving for the Sum of Distances A and B

Now, we can solve the equation for \((A + B)\):

\( (A + B)^2 + 40000 = 84100 \)

\( (A + B)^2 = 84100 - 40000 \)

\( (A + B)^2 = 44100 \)

To find \(A + B\), we take the square root of both sides:

\( A + B = \sqrt{44100} \)

\( A + B = 210 \)

The sum of the distances A and B that Maheep walked is 210 m.

Let's summarize the movements and resulting displacements:

The total horizontal displacement is \( (A+B) - 0 = A+B \). The total vertical displacement is \( -200 - 0 = -200 \), with a magnitude of 200.

Using the Pythagorean theorem: \( (A+B)^2 + (-200)^2 = 290^2 \).

\( (A+B)^2 + 40000 = 84100 \)

\( (A+B)^2 = 44100 \)

\( A+B = \sqrt{44100} = 210 \)

Revision Table: Key Concepts

Additional Information: Distance vs. Displacement

It's important to distinguish between distance and displacement, especially in problems involving multiple turns.

• Distance: This is a scalar quantity that represents the total length of the path traveled. If Maheep walked 10m North and then 10m South, the total distance is 20m.
• Displacement: This is a vector quantity representing the change in position from the starting point to the ending point. It is the length of the straight line connecting the start and end points, along with a direction. If Maheep walked 10m North and then 10m South back to his starting point, his total displacement is 0m.

In this problem, the distances A and B contribute to the total distance Maheep walked, but the shortest distance between his office and home is the magnitude of his displacement.

The problem cleverly uses the concept of displacement and the Pythagorean theorem by ensuring the net movement in one direction (horizontal) and the net movement in the perpendicular direction (vertical) form the legs of a right triangle whose hypotenuse is the shortest distance.

The final answer is the sum of the distances A and B, which is 210 m.