Point A is $30$ m to the North of point B. Point A is $10$ m to the west of point C. Point C is $20$m to the North of point D. Point E is $20$ m to the East of point D. Point F is $20$ m to the South of point E. What is the shortest distance from the point B to point E.
To find the shortest distance between point B and point E, we can use coordinate geometry. We need to determine the coordinates of both points based on the given relative positions.
Setting Up the Coordinate System
Let's establish a coordinate system. A convenient starting point is to place point B at the origin (0, 0).
- Point B = (0, 0)
Determining Coordinates of Other Points
Now, let's find the coordinates of the other points step-by-step:
Position of Point A relative to B:
Point A is 30 m to the North of point B. In a coordinate system where North is the positive y-direction, this means:
- Point A = (0, 0 + 30) = (0, 30)
Position of Point C relative to A:
Point A is 10 m to the West of point C. This means point C is 10 m to the East of point A. East corresponds to the positive x-direction.
- Point C = (0 + 10, 30) = (10, 30)
Position of Point D relative to C:
Point C is 20 m to the North of point D. This means point D is 20 m to the South of point C. South corresponds to the negative y-direction.
- Point D = (10, 30 - 20) = (10, 10)
Position of Point E relative to D:
Point E is 20 m to the East of point D. East corresponds to the positive x-direction.
- Point E = (10 + 20, 10) = (30, 10)
(Note: The information about point F is not needed to calculate the distance between B and E.)
Calculating the Shortest Distance from B to E
The shortest distance between two points in a plane is a straight line. We can calculate this using the distance formula, which is derived from the Pythagorean theorem. The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $Using the coordinates of B=(0, 0) and E=(30, 10):
- $x_1 = 0$, $y_1 = 0$
- $x_2 = 30$, $y_2 = 10$
Substitute these values into the distance formula:
$ d_{BE} = \sqrt{(30 - 0)^2 + (10 - 0)^2} $ $ d_{BE} = \sqrt{(30)^2 + (10)^2} $ $ d_{BE} = \sqrt{900 + 100} $ $ d_{BE} = \sqrt{1000} $To simplify the square root:
$ d_{BE} = \sqrt{100 \times 10} $ $ d_{BE} = \sqrt{100} \times \sqrt{10} $ $ d_{BE} = 10\sqrt{10} \, \text{m} $Summary Table
| Point | Coordinates (x, y) |
|---|---|
| B | (0, 0) |
| A | (0, 30) |
| C | (10, 30) |
| D | (10, 10) |
| E | (30, 10) |
The shortest distance calculated between point B and point E is $10\sqrt{10}$ meters.
Important Questions from Direction and Distance (Notes)
- If a map is placed in such a manner that east becomes southeast, then what will northeast become?
Rakesh left home and walked $5$km southwards, then turned right and walked $2$km and again turned right and walked $5$ km and finally again turned left and walked $5$ km. The shortest distance between the final position and home is.
- The door of Aman's house faces east. From the back side of his house, Aman walks straight 30m. Again he walks 20m after turning to his left, then he turns to his right and walks 30m. Finally he turn towards north and walks 20m and stops. So, what is the distance between the starting and the end point?
- Smitha runs 10 km south from her flat, turns left and walks 23 km again turns left and walks 40 km then turns right and walks 5 km to reach her office. In which direction is the office from her house?
I starts from Point A and drives 5 km towards the north. He then takes a right turn, drives 2 km, turns right and drives 9 km. He then takes a right turn and drives 7 km. He takes a final right turn, drives 4 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are $90^\circ$ turns only unless specified.)
