Understanding Malini's Journey and Directional Travel

The question asks for the shortest distance between Malini's office (starting point) and the bank (ending point). Malini's journey involves multiple turns and distances covered in different directions.

Let's break down her movement step by step, keeping track of her direction and the distance covered in each leg of the journey.

Step-by-Step Analysis of Malini's Path

• Starting Point: Malini's Office. She starts facing West.
• First Leg: She went 2 km straight. Since she was facing West, she traveled 2 km West.
• Second Leg: She turned to her right. If you are facing West, turning right means turning North. She went 3 km in this direction, so she traveled 3 km North.
• Third Leg: She turned left. If you are facing North, turning left means turning West. She walked 2 km in this direction, so she traveled 2 km West.
• Ending Point: The Bank.

Visualizing the Displacement

To find the shortest distance, we need to find the total displacement from the starting point (Office) to the ending point (Bank). Displacement is the straight-line distance and direction from the start to the end, regardless of the path taken.

Let's consider the directions and distances traveled:

• Total distance traveled West = 2 km (first leg) + 2 km (third leg) = 4 km West.
• Total distance traveled North = 3 km (second leg) = 3 km North.

Her net displacement is 4 km West and 3 km North from her office.

We can represent this displacement as a right-angled triangle, where:

• One side represents the total displacement West (4 km).
• Another side represents the total displacement North (3 km).
• The hypotenuse of this triangle represents the shortest distance between her office and the bank.

Calculating the Shortest Distance using the Pythagorean Theorem

The shortest distance is the hypotenuse of a right-angled triangle with sides of length 4 km and 3 km. We can use the Pythagorean theorem to calculate this distance.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (\(\text{c}\)) is equal to the sum of the squares of the other two sides (\(\text{a}\) and \(\text{b}\)).

Mathematically, this is written as:

\(\text{c}^2 = \text{a}^2 + \text{b}^2\)

In our case:

• \(\text{a}\) = total displacement West = 4 km
• \(\text{b}\) = total displacement North = 3 km
• \(\text{c}\) = shortest distance (hypotenuse)

Let's plug in the values:

\(\text{c}^2 = (4 \text{ km})^2 + (3 \text{ km})^2\)

\(\text{c}^2 = 16 \text{ km}^2 + 9 \text{ km}^2\)

\(\text{c}^2 = 25 \text{ km}^2\)

To find \(\text{c}\), we take the square root of both sides:

\(\text{c} = \sqrt{25 \text{ km}^2}\)

\(\text{c} = 5 \text{ km}\)

So, the shortest distance between Malini's office and the bank is 5 km.

Revision Table: Key Concepts

Additional Information: Direction and Displacement

In problems involving movement and direction, it's crucial to distinguish between distance traveled and displacement. Distance traveled is the sum of the lengths of all segments of the journey. Displacement, on the other hand, only cares about the initial and final positions.

When movements are along perpendicular directions (like North-South and East-West), the total displacement can often be found by treating the movements in each perpendicular direction separately and then using vector addition or the Pythagorean theorem, as we did in this problem.

For instance, moving 4 km West and 3 km North results in a displacement that is the hypotenuse of a 4 km by 3 km right triangle, giving a magnitude of 5 km.