Shortest distance can be found by connecting the initial point( home) to end in a way which forms a right angle triangle...
base=6
height=4.5
thus
hypotenous ( short. D.)=7.5(Pythagoras Theorem)

**Problem Analysis:**
In this problem, we are given that Dhruv walks 6 km on a straight road inclined at 45 degrees to his house. After that, he turns left 90 degrees and walks straight for 4.5 km to reach his office. We need to find the shortest distance between his house and his office.

**Solution:**
To solve this problem, let's consider the given information and analyze the situation step by step.

1. Initial Distance:
Dhruv walks 6 km on a straight road inclined at 45 degrees to his house. This means that he forms a right-angled triangle with the road as the hypotenuse and the distance between his house and the road as one of the legs. Since the road is inclined at 45 degrees, the triangle formed is an isosceles right-angled triangle. Therefore, the distance between his house and the road is equal to 6 km.

2. Turning Left:
After walking 6 km, Dhruv turns left 90 degrees. This means that he changes his direction and starts walking perpendicular to the initial road.

3. Walking Straight to Office:
After turning left, Dhruv walks straight for 4.5 km to reach his office.

4. Shortest Distance:
To find the shortest distance between his house and his office, we need to find the length of the hypotenuse of the right-angled triangle formed by the distance between his house and the road (6 km) and the distance he walks straight to reach his office (4.5 km). Using the Pythagorean theorem, we can calculate the length of the hypotenuse.

Hypotenuse = √(House distance^2 + Office distance^2)
= √(6^2 + 4.5^2)
= √(36 + 20.25)
= √56.25
= 7.5 km

Therefore, the shortest distance between his house and his office is 7.5 km.

**Answer:**
The correct answer is option D) 7.5 km.