MAT: Distance and Direction (With full Concepts)

Representation of Direction

This is basically the first step in solving questions related to distance and direction. In practice, we generally use 8 directions East, West, North, South, East-North, South-East, South-West, North-West. To explain it more clearly the following figure is presented

The angle between each of the main direction is 90. But the change in angle between East and North-East is 45°. For example, The angle between East and North is 90° also East and South is 90°. 

Points to Remember:

• For a person facing towards East, has North in left side and South on his right side.
• For a person facing towards North has, has West on his left and East on his right.
• For a person facing towards West, has South on his left and North on his right.
• Lastly, for a person facing towards South, has East on his left and West on his right.

Clockwise and Anti-clockwise Direction:

The direction similar to that of a watch(clock) is termed as clockwise direction. While the opposite of the direction of watch is known as anti-clockwise direction. 

Note: When a person is facing in east direction, then after turning 90° in a clockwise direction, he is now facing in south direction. And for the same person if had turned 90°in anticlockwise direction he would have facing North direction. 

Clockwise direction is also known as a rotation through a positive 90° (+90°). And the anti-clockwise direction is termed as a rotation through negative 90°(-90°).

Pythagoras Theorem

In a right-angled triangle, the sum of the square of Base and perpendicular is equal to the sum of the hypotenuse.

In the given triangle ABC, AB is perpendicular, BC is base and AC is the hypotenuse. Then, by the Pythagorean theorem,
AB^2 +BC^2=AC^2

Q.N. 1  A father and a son started their journey from the same place. Father walked 120km South, then turned left and moved 80km. Son moved 30km East from the origin point and got an accident. What is the minimum distance father should travel in order to come for help?

Solution:

Let O be the point of origin, after walking 120km East suppose he reaches to point A, then after turning left he reached point C. Son also started his journey from point O and reached to D.

From figure,

  OA = DB = 120km

  AC = 80km

  Therefore, BC = (80 -  30)km

  DC = ?

Now, In right-angled triangle DBC,

    Using Pythagorean theorem we have

          DC = √ (DB² + BC²)

   or,  DC = √ (120² + 50²)

   or,  DC = √ (16900)

  ∴ DC = 130 km

Hence, The minimum distance that the father should travel in order to help his son is 130km.

 

Q.N. 2 A disoriented foreigner walked 90km straight East from a point, then turned right and moved 40km then again took left and walked 50km. Lastly, he took left again and moved for 70km. What is the minimum distance he should cover to reach the origin point?

Solution:

Let A be the origin point from where the foreigner has started his journey. After walking 90km let he reaches to point B, then after turning right, he reached to point C, then to point D. Finally, he reached to point F. 

From figure,

AB = 90 km

FD = 70 km

BC = ED = 40 km

DC = EB = 50 km

 ∴ AE = AB - EB

          = (90 - 50) km

           =  40 km

and EF = FD - ED

            = (70 - 40) km

            = 30 km

In right-angled triangle AEF

 Using Pythagorean theorem,

      AF = √ (AE² + EF²)

   or,  AF = √ (40² + 30²)

   or,  AF = √ (2500)

  ∴ AF = 50 km

Hence, The minimum distance he should travel to reach the origin point is 50 km.

.

Note:- When a person turns to his right side, then the movement is in a clockwise direction and when a person turns to left then the movement is anti-clockwise. 

Questions for practice

1.Rahul walks 40 meters towards the south then turns to his right and starts walking straight till he completes another 40 meters. Then again turning to his left he walks 10 metres. He then turns to his left and walks for 40 metres. How far is he from his initial position?

A) 80 km
B) 50 km
C) 70 km
D) 60 km

Answer is B)