Rahman went from his office to the district headquarters. He started his journey facing west. First, he went 20 km straight; then he turned to his left and went 9 km; finally, he turned right and went 20 km to reach the district headquarters.
41 km
Understanding Rahman's Journey and Finding Shortest Distance
The question asks us to find the shortest distance between Rahman's office (start point) and the district headquarters (end point) based on his specific path. The shortest distance between two points is always a straight line.
Mapping Rahman's Path
Rahman starts at his office, facing west. We can break down his journey into steps:
- He goes 20 km straight, which is 20 km west from his office.
- He turns to his left. Since he was facing west, turning left means he is now facing south. He goes 9 km in this direction, so 9 km south.
- Finally, he turns right. Since he was facing south, turning right means he is now facing west. He goes 20 km in this direction, so 20 km west.
Calculating Total Displacement
To find the shortest distance, we need to determine the total displacement from the starting point (office). Displacement is the straight-line distance and direction from the start to the end point.
Let's look at the movements in each direction:
- Westward movement: Rahman first went 20 km west and then another 20 km west. The total westward displacement is $20 \text{ km} + 20 \text{ km} = 40 \text{ km}$.
- Southward movement: Rahman went 9 km south. The total southward displacement is $9 \text{ km}$.
His final position relative to his office is 40 km west and 9 km south.
Applying the Pythagorean Theorem
The total displacement forms a right-angled triangle where:
- One leg is the total westward displacement (40 km).
- The other leg is the total southward displacement (9 km).
- The hypotenuse is the shortest distance between the office and the district headquarters.
We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). The formula is $a^2 + b^2 = c^2$.
Let $a = 40$ km and $b = 9$ km. We need to find $c$.
The calculation is as follows:
$c^2 = (40 \text{ km})^2 + (9 \text{ km})^2$
$c^2 = 1600 \text{ km}^2 + 81 \text{ km}^2$
$c^2 = 1681 \text{ km}^2$
$c = \sqrt{1681 \text{ km}^2}$
$c = 41 \text{ km}$
The shortest distance between Rahman's office and the district headquarters is 41 km.
Revision Table: Key Concepts
| Concept | Description | Relevance Here |
|---|---|---|
| Shortest Distance | The straight-line distance between two points. Also known as the magnitude of displacement. | What the question asks for. |
| Displacement | The overall change in position from start to end, including direction. | Calculated by summing movements in perpendicular directions (West/East and North/South). |
| Pythagorean Theorem | In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. | Used to find the shortest distance (hypotenuse) from the total horizontal and vertical displacements. |
Additional Information: Distance vs. Displacement
It's important to distinguish between distance and displacement:
- Distance: The total length of the path traveled. In this case, the total distance Rahman traveled is $20 \text{ km} + 9 \text{ km} + 20 \text{ km} = 49 \text{ km}$.
- Displacement: The straight-line distance from the starting point to the ending point, along with the direction. We calculated the magnitude of displacement to be 41 km.
The question specifically asks for the "shortest distance," which refers to the magnitude of the displacement.
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