Manish is facing South. He took 90° right and walked 8 km. then he turn right and walked 6 km. What is the minimum distance between starting point to ending point?
10 km
Solving the Direction and Distance Problem
This question asks us to find the minimum distance between Manish's starting point and his ending point after a series of movements based on directions and turns.
Understanding the Movements
Let's break down Manish's path step by step:
- Initially, Manish is facing South.
- He takes a 90° right turn. If you are facing South, a right turn means turning towards the West. So, he is now facing West.
- He walks 8 km in the West direction. Let's assume his starting point is the origin (0,0) on a coordinate plane. Walking 8 km West takes him to the point (-8, 0).
- From this point, he turns right again. He is currently facing West. A right turn from West means turning towards the North. So, he is now facing North.
- He walks 6 km in the North direction. From the point (-8, 0), walking 6 km North takes him to the point (-8, 6).
Determining Start and End Points
- Starting Point: (0, 0)
- Ending Point: (-8, 6)
The minimum distance between the starting point and the ending point is the straight-line distance, which is the displacement.
Calculating Minimum Distance using Pythagorean Theorem
The path taken by Manish (8 km West followed by 6 km North) forms two sides of a right-angled triangle. The 8 km movement is along the West-East axis, and the 6 km movement is along the North-South axis. These two paths are perpendicular to each other, forming the two shorter sides (legs) of a right-angled triangle. The minimum distance between the start and end points is the hypotenuse of this triangle.
We can use the Pythagorean theorem to find the length of the hypotenuse (\(c\)), given the lengths of the two legs (\(a\) and \(b\)). The theorem states:
\[c^2 = a^2 + b^2\]
In this case:
- Leg a = 8 km (distance moved West)
- Leg b = 6 km (distance moved North)
- Hypotenuse c = Minimum distance between start and end points
Substituting the values into the theorem:
\[c^2 = (8 \, \text{km})^2 + (6 \, \text{km})^2\]
\[c^2 = 64 \, \text{km}^2 + 36 \, \text{km}^2\]
\[c^2 = 100 \, \text{km}^2\]
To find \(c\), we take the square root of both sides:
\[c = \sqrt{100 \, \text{km}^2}\]
\[c = 10 \, \text{km}\]
So, the minimum distance between Manish's starting point and ending point is 10 km.
Summary of the Path
| Action | Direction/Turn | Distance | New Position (relative to Start) |
|---|---|---|---|
| Starting | Facing South | 0 km | (0,0) |
| 1st Movement | 90° Right (towards West) | 8 km | (-8, 0) |
| 2nd Movement | Right Turn (towards North) | 6 km | (-8, 6) |
The straight line from (0,0) to (-8,6) is the minimum distance, which is 10 km.
Revision Table: Key Concepts in Direction and Distance
| Concept | Explanation | How it Applied Here |
|---|---|---|
| Cardinal Directions | North, South, East, West. Used as primary reference points. | Movements were West and North. |
| Turns (Right/Left) | Right turn is 90° clockwise. Left turn is 90° counter-clockwise. | Manish took 90° right turns from South (to West) and from West (to North). |
| Minimum Distance (Displacement) | The shortest straight-line distance between the initial and final positions. | Calculated as the hypotenuse of the right-angled triangle formed by the movements. |
| Pythagorean Theorem | In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (\(a^2 + b^2 = c^2\)). | Used to find the minimum distance (hypotenuse) from the two perpendicular movements (legs). |
Additional Information: Directions and Geometry
Problems involving direction and distance often require visualizing movements and using basic geometry, particularly the Pythagorean theorem, to find the shortest path or displacement.
- Visualisation: Always try to draw a diagram or mentally trace the path taken. Starting at an origin point (like 0,0) and using coordinates can be very helpful, especially when movements are along the cardinal directions.
- Perpendicular Movements: When movements are perpendicular (like North-South and East-West), they naturally form the legs of a right-angled triangle with the displacement as the hypotenuse. This is where the Pythagorean theorem is crucial for finding the minimum distance.
- Understanding Turns: A standard "right turn" or "left turn" usually implies a 90° turn unless specified otherwise. Knowing which direction results from a turn from a given facing direction is fundamental (e.g., Right from South is West, Right from West is North).
- Distance vs. Displacement: The total distance walked is the sum of the lengths of all segments of the path (8 km + 6 km = 14 km in this case). The minimum distance or displacement is the straight-line distance from the start to the end point (10 km here). The question specifically asks for the minimum distance.
Important Questions from Direction and Distance
- Vijendra walks a certain distance, say X metres, from his home towards the west. Then he turns left and walks 23 metres. After that, he turns left and walks 36 metres. Then he turns left again to walk 23 metres. He finally turns left and walks 18 metres to reach his home. Find the value of X.
Gaurav exits from the backdoor of his north-facing house and walks 25 m straight, then he takes a left turn and walks 36 m, then he turns left and walks 47 m. He turns left again and walks 36 m. How far and in which direction is he from his house now?
Reeta is standing facing south-east. First, she turns 135° clockwise. After that, she turns 90 ° anticlockwise. Then she turns 45 ° clockwise, followed by a 135 ° anticlockwise turn. In which direction is she facing now?
Sahasra started running from her house towards the north. After 40 meters she turned left and ran for 75 meters. She then turned right and ran for 30 meters, and again turned right to run 75 meters. In which direction was she running finally?
At the time of sunset, Lopa and Kritika are sitting facing each other. If the shadow of Lopa falls to the right of Kritika, in which direction is Kritika facing?
