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)
- A woman leaves her home. She walks 40 m in North-West direction and then 90 m in South-East direction. Then, she moves 30 m in North direction. How far is she now from her initial position ?
- N sits to the North of L and West of M. L sits to the East of O. If everyone is facing North, which direction of O, is N?
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.)
A tourist drives $20 \text{ km}$ towards east, turns right and drives $6 \text{ km}$, then drives $6 \text{ km}$ towards west. He then turns to his left and drives $4 \text{ km}$ and finally turns right and drives $14 \text{ km}$. Where is he from his starting point?
- A Couple started to their offices in the morning from their home. While husband drives 3 km South and then 2 km to his right to reach his office, wife goes 2 km towards East for her office. With respect to husband's office, how far and in which direction is the wife's office ?
