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 answer is $\sqrt{569}$km.

To find the shortest distance between point A and point I, we can set up a Cartesian coordinate system. Let point A be the origin with coordinates $\left(0,0\right)$.

Step 1: Determine the coordinates of Rohit's path starting from point A (0, 0):
- Rohit starts at point A $\left(0,0\right)$.
- He moves 4 km towards the north to reach point B. Thus, the coordinates of B are:
$B=\left(0,4\right)$
- From B, he turns left (west) and moves 3 km to reach point C. Thus, the coordinates of C are:
$C=\left(0-3,4\right)=\left(-3,4\right)$
- From C (facing west), he turns left (south) and moves 5 km to reach point D. Thus, the coordinates of D are:
$D=\left(-3,4-5\right)=\left(-3,-1\right)$
- From D (facing south), he turns right (west) and moves 8 km to reach point E. Thus, the coordinates of E are:
$E=\left(-3-8,-1\right)=\left(-11,-1\right)$

Step 2: Determine the coordinates of Sonia's path starting from point E (-11, -1):
- Sonia starts at point E $\left(-11,-1\right)$.
- She moves 5 km towards the south to reach point F. Thus, the coordinates of F are:
$F=\left(-11,-1-5\right)=\left(-11,-6\right)$
- From F (facing south), she turns right (west) and moves 6 km to reach point G. Thus, the coordinates of G are:
$G=\left(-11-6,-6\right)=\left(-17,-6\right)$
- From G (facing west), she turns left (south) and moves 7 km to reach point H. Thus, the coordinates of H are:
$H=\left(-17,-6-7\right)=\left(-17,-13\right)$
- From H (facing south), she turns right (west) and moves 3 km to reach point I. Thus, the coordinates of I are:
$I=\left(-17-3,-13\right)=\left(-20,-13\right)$

Step 3: Calculate the shortest distance between point A and point I:
Using the distance formula between point A $\left(0,0\right)$ and point I $\left(-20,-13\right)$:
$\text{Distance}=\sqrt{{\left(x_A-x_I\right)}^2+{\left(y_A-y_I\right)}^2}$
Substituting the coordinates:
$\text{Distance}=\sqrt{{\left(0-\left(-20\right)\right)}^2+{\left(0-\left(-13\right)\right)}^2}$
$\text{Distance}=\sqrt{{20}^2+{13}^2}$
$\text{Distance}=\sqrt{400+169}=\sqrt{569}\text{km}$
Therefore, the shortest distance between point A and point I is indeed $\sqrt{569}$ km.