an airplane flies 200km from city a to city b east,then 200km south from city b to c,and finally 100km northwest to city d.how far is it from city a to city d.in what direction must the airplane head to return directly to city a from city d?

location of c is (200,-200)

location of d
x = 200 - 100 cos 45
y = -200 + 100 sin 45

keep heading NW obviously
distance = a to d:

= 200 sqrt 2 - 100

To find the distance from city A to city D, we need to calculate the total distance traveled by the airplane.

The airplane flies 200 km from city A to city B east.
Then it flies another 200 km south from city B to city C.
Finally, it flies 100 km northwest to city D.

We can visualize the movement of the airplane as follows:

B
A--C
D

To find the distance from city A to city D, we can draw a right-angled triangle with sides representing the distances traveled in each direction.

The distance traveled east-west (horizontal distance) is 200 km + 0 km (no horizontal movement).

The distance traveled north-south (vertical distance) is 0 km + 200 km.

Using Pythagoras' theorem (a^2 + b^2 = c^2) to find the hypotenuse (total distance traveled), we have:

Total distance = √[(horizontal distance)^2 + (vertical distance)^2]
= √[(200 km)^2 + (200 km)^2]
= √[40000 km^2 + 40000 km^2]
= √(80000 km^2)
= 282.8 km (approx.)

Therefore, the distance from city A to city D is approximately 282.8 km.

To return directly to city A from city D, the airplane must head in the opposite direction of the previous northwest movement. In this case, it should head southeast.

To determine the total distance from City A to City D, we need to break down the journey into its individual components and calculate them separately.

1. The distance from City A to City B is given as 200 km.
2. The distance from City B to City C is given as 200 km.
3. The distance from City C to City D is given as 100 km.

To calculate the distance from City A to City D, we need to add the distances of the three segments:

Distance from A to D = (Distance from A to B) + (Distance from B to C) + (Distance from C to D)

Distance from A to D = 200 km + 200 km + 100 km

Distance from A to D = 500 km

Therefore, the distance from City A to City D is 500 km.

To determine the direction the airplane must head to return directly to City A from City D, we need to consider the relative position of the two cities.

City D is northwest of City A. To return directly to City A, the airplane must head in the opposite direction, which is southeast.

So, the airplane must head southeast to return directly to City A from City D.