I travel 20 miles towards north and then travel 25 miles eastward. I then travel 40 miles rightwards, then travel 30 miles towards left and then travels 12 miles to the left and finally 20 miles northwards. How far am i pproximately from my original destination and in what direction ?

Let's break down the journey:

1. You start by traveling 20 miles north.
2. Then you travel 25 miles east.
3. "Traveling rightwards" depends on the direction you're facing. Since you were traveling east, turning right puts you facing south. So you travel 40 miles south.
4. "Traveling left" from a southern direction puts you facing east. So you travel 30 miles east.
5. "Traveling left" from an eastern direction puts you facing north. So you travel 12 miles north.
6. Finally, you travel another 20 miles north.

Now, let's calculate the total distance traveled in each cardinal direction:

• North: 20 miles + 12 miles + 20 miles = 52 miles
• East: 25 miles + 30 miles = 55 miles
• South: 40 miles
• West: 0 miles

So, your final position relative to your starting point is 52 miles - 40 miles = 12 miles to the north and 55 miles to the east.

To find the straight-line distance from your starting point, you can use the Pythagorean theorem, which states that the square of the length of the hypotenuse (the straight-line distance in this case) is equal to the sum of the squares of the other two sides. In this case, the other two sides are the distances traveled north and east, which are 12 miles and 55 miles respectively.

So, the straight-line distance is sqrt((12 miles)^2 + (55 miles)^2) = sqrt(144 + 3025) = sqrt(3169) = approximately 56.3 miles.

The direction from your starting point to your final position is the angle whose tangent is the ratio of the distance traveled north to the distance traveled east, or tan^-1(12/55). This is approximately 12.4 degrees north of east.