Directions (38-40): Study the information carefully and answer the questions given below.

Rohit is standing at point A and wants to reach point E. He can only move in a straight line, but there are obstacles in his path. Rohit first moves 4 km towards the north and reaches point B. From there, he turns left and moves 3 km towards the west and reaches point C. He then turns left again and moves 5 km towards the south and reaches point D. Finally, he turns right and moves 8 km and reaches point E. Sonia wants to travel from point E to point A, but there is an additional obstacle in her path. There is a mountain between points E and A, which she cannot cross. Sonia first moves 5 km towards the south from point E and reaches point F. From there, she turns right and moves 6 km and reaches point G. She then turns left and moves 7 km towards the south and reaches point H. Finally, she turns right and moves 3 km and reaches point I. From point I, Sonia needs to reach point A, with shortest distance.

What is the shortest distance between point A and point I?

The correct option is $\sqrt{569}$km.

To find the shortest distance between point A and point I, we can determine the coordinates of each point by setting point A as the origin (0, 0) on a standard Cartesian coordinate plane where:
- The positive y-axis represents North (N)
- The negative y-axis represents South (S)
- The positive x-axis represents East (E)
- The negative x-axis represents West (W)

Step 1: Track Rohit's path to find the coordinates of point E
1. Rohit starts at point A:
$A=(0,0)$
2. He moves 4 km North to reach point B:
$B=(0,4)$
3. From B, he turns left (West) and moves 3 km to reach point C:
$C=(0-3,4)=(-3,4)$
4. From C, he turns left (South) and moves 5 km to reach point D:
$D=(-3,4-5)=(-3,-1)$
5. From D, he turns right (West, since he was going South) and moves 8 km to reach point E:
$E=(-3-8,-1)=(-11,-1)$

Step 2: Track Sonia's path starting from point E to find the coordinates of point I
1. Sonia starts at point E:
$E=(-11,-1)$
2. She moves 5 km South to reach point F:
$F=(-11,-1-5)=(-11,-6)$
3. From F, she turns right (West, since she was going South) and moves 6 km to reach point G:
$G=(-11-6,-6)=(-17,-6)$
4. From G, she turns left (South, since she was going West) and moves 7 km South to reach point H:
$H=(-17,-6-7)=(-17,-13)$
5. From H, she turns right (West, since she was going South) and moves 3 km to reach point I:
$I=(-17-3,-13)=(-20,-13)$

Step 3: Calculate the shortest distance between point A and point I
The coordinates are A(0, 0) and I(-20, -13). Using the distance formula:
$\text{Distance}=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
Substituting the coordinates:
$AI=\sqrt{{(-20-0)}^2+{(-13-0)}^2}$
$AI=\sqrt{{(-20)}^2+{(-13)}^2}$
$AI=\sqrt{400+169}$
$AI=\sqrt{569}\text{km}$