Calculating the Shortest Distance for a Journey

The question asks for the shortest distance between the start and end point of a journey where a person walks 3 km towards the west and then turns to travel 4 km towards the north.

Let's visualize the journey:

• The person starts at a point (let's call it Start).
• They walk 3 km west from Start to a turning point (let's call it Point A). This path is a straight line segment 3 km long directed west.
• From Point A, they turn and walk 4 km north to the end point (let's call it End). This path is a straight line segment 4 km long directed north.

Since walking west and then turning to walk north involves movements perpendicular to each other (west is perpendicular to north), the path taken forms two sides of a right-angled triangle. The starting point, the turning point (Point A), and the ending point form the vertices of this right-angled triangle.

The two sides that are perpendicular are the 3 km west distance and the 4 km north distance. These are the two legs (or cathetus) of the right-angled triangle. The shortest distance between the start point and the end point is a straight line connecting these two points. This straight line forms the hypotenuse of the right-angled triangle.

To find the length of the hypotenuse (the shortest distance), we can use the Pythagorean Theorem.

Using the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is written as:

\(a^2 + b^2 = c^2\)

In our case:

• Side a = 3 km (distance west)
• Side b = 4 km (distance north)
• Side c = Shortest distance between start and end (hypotenuse)

Step-by-Step Calculation

1. Identify the lengths of the two perpendicular sides (legs of the right triangle):
   \(a = 3 \text{ km}\)
   \(b = 4 \text{ km}\)
2. Apply the Pythagorean Theorem:
   \(c^2 = a^2 + b^2\)
3. Substitute the values of a and b into the equation:
   \(c^2 = (3 \text{ km})^2 + (4 \text{ km})^2\)
4. Calculate the squares of the sides:
   \(c^2 = 9 \text{ km}^2 + 16 \text{ km}^2\)
5. Sum the squared values:
   \(c^2 = 25 \text{ km}^2\)
6. Find the square root of \(c^2\) to get the value of c:
   \(c = \sqrt{25 \text{ km}^2}\)
   \(c = 5 \text{ km}\)

Therefore, the shortest distance between the start and the end point of X's journey is 5 km.

Revision Table: Key Concepts for Distance Calculation

Additional Information: Distance vs. Displacement

It is important to understand the difference between total distance traveled and displacement (shortest distance).

• Total Distance Traveled: This is the sum of the lengths of all the segments of the path taken. In this journey, the total distance traveled is 3 km + 4 km = 7 km.
• Displacement: This is a vector quantity representing the straight-line distance and direction from the initial position to the final position. The magnitude of the displacement vector is the shortest distance between the start and end points. In this case, the magnitude of the displacement is 5 km, and its direction would be south-east relative to the turning point (Point A), or a specific angle relative to the start point. The question only asks for the shortest distance, which is the magnitude of the displacement.

When a person travels in directions that are perpendicular (like west and north), the path forms a right angle, and the shortest distance (displacement magnitude) can be found using the Pythagorean theorem.