Mridul cycles 145 m towards the east, and then turns left and cycles 155 m. Again, he turns left and cycles 225 m. He finally turns left and cycles 95 m to reach his home. What is the shortest distance between his home and the starting point?
100 m
Mridul's Cycling Path: Shortest Distance Calculation
This problem involves finding the shortest possible distance between Mridul's starting point and his home after he completes a journey involving several turns. We need to calculate his net displacement.
Mridul's Movements: East, North, West, South Tracking
Let's follow Mridul's journey:
- He begins by cycling 145 m towards the east.
- He then turns left and cycles 155 m (now moving North).
- He turns left again and cycles 225 m (now moving West).
- Finally, he turns left once more and cycles 95 m (now moving South) to arrive at his home.
Net Displacement: East-West and North-South Components
To determine the shortest distance, we calculate the net effect of his movements along the cardinal directions (East-West and North-South).
East-West Displacement: Mridul first travels 145 m East (+145 m). Then he travels 225 m West (-225 m). The net displacement in the East-West direction is calculated as: \( \Delta x = 145 \, \text{m} - 225 \, \text{m} = -80 \, \text{m} \) A negative value means the net movement is Westward. So, his final position is 80 m West of his starting point.
North-South Displacement: Next, he travels 155 m North (+155 m). Then he travels 95 m South (-95 m). The net displacement in the North-South direction is calculated as: \( \Delta y = 155 \, \text{m} - 95 \, \text{m} = 60 \, \text{m} \) A positive value means the net movement is Northward. So, his final position is 60 m North of his starting point.
Therefore, Mridul's home is located 80 m West and 60 m North relative to his starting position.
Shortest Distance Geometry: Pythagoras Theorem Application
The shortest distance is a straight line from the start to the end point. This forms the hypotenuse of a right-angled triangle, where the legs are the net displacements along the West and North directions.
- Westward displacement = \( 80 \, \text{m} \)
- Northward displacement = \( 60 \, \text{m} \)
We use the Pythagorean theorem (\( a^2 + b^2 = c^2 \)), where '\(a\)' and '\(b\)' are the lengths of the legs and '\(c\)' is the length of the hypotenuse (the shortest distance).
Step-by-Step Calculation:
- Define the lengths of the triangle's legs based on the net displacements:
- Westward leg (\(a\)) = \( 80 \, \text{m} \)
- Northward leg (\(b\)) = \( 60 \, \text{m} \)
- Apply the Pythagorean theorem: \( \text{Shortest Distance}^2 = a^2 + b^2 \).
- Substitute the values: \( \text{Shortest Distance}^2 = (80)^2 + (60)^2 \).
- Calculate the squares: \( \text{Shortest Distance}^2 = 6400 + 3600 \).
- Sum the squared values: \( \text{Shortest Distance}^2 = 10000 \).
- Find the shortest distance by taking the square root: \( \text{Shortest Distance} = \sqrt{10000} \).
- Calculate the final result: \( \text{Shortest Distance} = 100 \, \text{m} \).
Final Distance: Mridul's Home to Starting Point
The calculation shows that the shortest distance between Mridul's home and his original starting point is 100 m.
Similar Questions
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Important Questions from Direction and Distance Turns
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A woman runs 12 km towards her North, then 6 km towards her South and then 8 km towards her East. In which direction is she from her starting point ?
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Shekhar starts walking south from his school and then took one right turn, one left turn and one right turn to reach his shop. In which direction is Shekhar with respect to his school?
