Location of B is north of A and location of C is east of A. The distances AB and AC are 5 km and 12 km respectively. The shortest distance (in km) between the locations B and C is
UPSC - 2014
The problem describes the relative positions of three locations A, B, and C, and the distances between A and B, and A and C. We need to find the shortest distance between B and C. This scenario forms a right-angled triangle, where the shortest distance between B and C will be the hypotenuse. We can use the Pythagorean theorem to solve this.
Understand the relative positions:
This arrangement implies that the lines AB and AC are perpendicular to each other, forming a right angle at A.
Identify the given distances:
Formulate the geometric shape:
Since AB is along the North direction and AC is along the East direction from A, the triangle ABC is a right-angled triangle with the right angle at A.
Apply the Pythagorean theorem:
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. Here, BC is the hypotenuse.
\[ BC^2 = AB^2 + AC^2 \]Substitute the given values:
\[ BC^2 = (5 \text{ km})^2 + (12 \text{ km})^2 \]\[ BC^2 = 25 \text{ km}^2 + 144 \text{ km}^2 \]\[ BC^2 = 169 \text{ km}^2 \]Calculate the shortest distance BC:
Take the square root of both sides:
\[ BC = \sqrt{169 \text{ km}^2} \]\[ BC = 13 \text{ km} \]B) 13 km is the shortest distance between locations B and C, calculated using the Pythagorean theorem.
Two friends X and Y start running and they run together for 50 m in the same direction and reach a point. X turns right and runs 60 m, while y turns left and runs 40 m. Then X turns left and runs 50 m and stops, while Y turns right and runs 50 m and then stops. How far are the two friends from each other now?
UPSC - 2022
The problem describes the movement of two friends, X and Y, in a sequence of turns and distances. We need to determine the final distance between them. This can be solved by visualizing their paths on a 2D plane or by breaking down their movements into components.
X is at C (60, 100) and Y is at E (-40, 100).
Since both X and Y are at the same Y-coordinate (100), the distance between them is the absolute difference of their X-coordinates.
\[ \text{Distance} = |X_{\text{coordinate of X}} - X_{\text{coordinate of Y}}| \]\[ \text{Distance} = |60 - (-40)| \]\[ \text{Distance} = |60 + 40| \]\[ \text{Distance} = |100| \]\[ \text{Distance} = 100 \text{ m} \]A) 100m. The final positions of X and Y are (60, 100) and (-40, 100) respectively. The horizontal distance between them is \(60 - (-40) = 100\) m.
P, Q and R are three towns. The distance between P and Q is 60 km, whereas the distance between P and R is 80 km. Q is in the West of P and R is in the South of P. What is the distance between Q and R?
UPSC - 2019
This problem involves understanding directions and distances, which can be visualized as a right-angled triangle. We can use the Pythagorean theorem to find the distance between Q and R.
Correct Option: D) 100 Km
'A' started from his house and walked 20 m towards East, where his friend 'B' joined him. They together walked 10 m in the same direction. Then 'A' turned left while 'B' turned. right and travelled 2 m and 8 m respectively. Again 'B' turned left to travel 4 m followed by 5 m to his right to reach His office. 'A' turned right and travelled 12 m to reach his office. What is the shortest distance between the two
UPSC - 2019
This problem involves tracking the movements of two individuals, 'A' and 'B', using directions and distances. We need to determine the final positions of both 'A' and 'B' relative to their starting point and then calculate the shortest distance between their offices using the Pythagorean theorem.
We need to find the shortest distance between Office A \((42, 2)\) and Office B \((34, -13)\). We can use the distance formula:
\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the coordinates:
\[\text{Distance} = \sqrt{(34 - 42)^2 + (-13 - 2)^2}\] \[\text{Distance} = \sqrt{(-8)^2 + (-15)^2}\] \[\text{Distance} = \sqrt{64 + 225}\] \[\text{Distance} = \sqrt{289}\] \[\text{Distance} = 17 \text{ m}\]Correct Option: B) The shortest distance between the two offices is 17 m.
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?
UPSC - 2021
An angle less than 45° South of East
An angle less than 45° North of East
An angle more than 45° south of East
An angle more than 45° North of East
To find the final direction from the starting point, we need to determine the net displacement in the North-South direction and the East-West direction. We can then use trigonometry to find the angle and direction.
Shahid and Rohit start from the same point in opposite directions. After each 1 km, Shahid always turns left and Rohit always turns right. Which of the following statements is correct?
UPSC - 2015
After both have travelled 2 km, the distance between them is 4 km.
they meet after each has travelled 3 km.
They meet for the first time after each has travelled 4 km.
They go on without ever meeting again.
This problem involves tracking the movement of two individuals, Shahid and Rohit, who start from the same point and move in opposite directions, making turns after every 1 km. We need to determine when and if they meet, or the distance between them at certain points. Visualizing their paths on a grid or coordinate plane is the most effective approach.
B) they meet after each has travelled 3 km. As shown in the step-by-step analysis, after Shahid has travelled 3 km, he is at coordinates (0,1). After Rohit has travelled 3 km, he is also at coordinates (0,1). Therefore, they meet at this point.
