If an airplane flies 100 miles due north, then turns and flies 100 miles directly northeast, how far is it the?

The best way to do this is to break the question into parts. First the due north, then the northeast. Do them separately and then add them together.

Assume the plane starts at the origin of a graph. Flying due north 100 miles the plane will be sitting at (0, 100).

Start again from the origin. Flying 100 miles directly northeast, the plane is travelling at a 45 degree angle to the x or y axis. Using trig we can calculate the distance travelled up along the y-axis and across the x-axis.

Drawing out the triangle you'll see that to find the distance across the x-axis you need to find the adjacent, and you're given the hypotenuse and an angle. This means you need to use Cosine.

Cos(theta) = adjacent/hypotenuse.

Rearrange you get hypotenuse * Cos(theta) = adjacent.

100 * Cos(45) = 70.71 units along the x-axis.

Similarly you will find that the plane also travelled 70.71 units up the y-axis. (turn the graph sideways, again you need to find the adjacent)

You can check that this is right by substituting into the pythagorean theorem

a^2 + b^2 = c^2

70.71^2 + 70.71^2 = 100^2

Now you can add these coordinates together with the coordinates from part 1. (0,100) + (70.71, 70.71) = (70.71, 170.71)

Now we can use the pythagorean theorem again to find the straight line distance.

70.71^2 + 170.71^2 = c^2

34141.8082 = c^2

sqrt(34141.8082) = c

c = 184.78 = 185 miles

Hope this helps

Cheers