Calculating Shortest Distance in Direction Problems

This problem involves calculating the shortest distance between two points after a series of movements in different directions. We can visualize the movement as forming a right-angled triangle, where the initial and final positions are connected by the hypotenuse.

Step-by-Step Movement Analysis

Let's break down Mr. X's journey:

1. Mr. X starts at point ‘A’.
2. He travels 80 km towards the west. Let's call the point he reaches after this travel point ‘C’. So, the distance AC is 80 km, and the direction is West from A.
3. From point ‘C’, he takes a left turn. If he was going West, a left turn means turning towards South.
4. He travels 50 km after taking the left turn (South) and reaches point ‘B’. So, the distance CB is 50 km, and the direction is South from C.

Visualizing the Path and Shortest Distance

We can represent this journey using a diagram or by considering coordinates. If we start at A(0,0):

• Traveling West 80 km takes us to C(-80, 0).
• From C(-80, 0), traveling South 50 km takes us to B(-80, -50).

The points A, C, and B form a right-angled triangle, with the right angle at point C. Point A is the starting point, and point B is the ending point. The shortest distance between points ‘A’ and ‘B’ is the length of the straight line segment connecting A and B, which is the hypotenuse of the triangle ACB.

The lengths of the two sides forming the right angle are:

• AC (Westward distance) = 80 km
• CB (Southward distance) = 50 km

Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our case, AB is the hypotenuse.

Using the Pythagorean theorem:

\(AB^2 = AC^2 + CB^2\)

Substitute the distances:

\(AB^2 = (80 \, \text{km})^2 + (50 \, \text{km})^2\)

\(AB^2 = 6400 \, \text{km}^2 + 2500 \, \text{km}^2\)

\(AB^2 = 8900 \, \text{km}^2\)

To find the shortest distance AB, we take the square root of \(AB^2\):

\(AB = \sqrt{8900} \, \text{km}\)

We can simplify the square root:

\(\sqrt{8900} = \sqrt{100 \times 89}\)

\(\sqrt{8900} = \sqrt{100} \times \sqrt{89}\)

\(\sqrt{8900} = 10 \times \sqrt{89}\)

So, the shortest distance between points ‘A’ and ‘B’ is \(10\sqrt{89}\) km.

Comparing with Given Options

Let's compare our calculated shortest distance with the given options:

• Option 1: \(10\sqrt {93}\)
• Option 2: \(10\sqrt {39}\)
• Option 3: \(10\sqrt {98}\)
• Option 4: \(10\sqrt {89}\)

Our calculated shortest distance, \(10\sqrt{89}\) km, matches Option 4.

Revision Table: Key Concepts

Additional Information: Understanding Direction Problems

Direction and distance problems often involve plotting movements on a 2D plane. The shortest distance between two points is always a straight line. When movements involve turns at right angles (like North then East, or West then South), the problem can usually be solved using the Pythagorean theorem, as the paths form right-angled triangles.

Key points to remember:

• Visualize the path or sketch a simple diagram.
• Identify the starting and ending points.
• If movements form right angles, look for right-angled triangles.
• Use the Pythagorean theorem to find the shortest distance (hypotenuse) if a right triangle is formed.
• Pay attention to left and right turns relative to the current direction of travel.

For example, from North, a left turn is West, and a right turn is East. From West, a left turn is South, and a right turn is North, and so on.