A boat leaves the shore and travels 25km east and then 10km,60 degree east of north .can you use the phytagorean theory in finding the boats resultant?

The Pythagorean Theorem does come into play in this, but you first need to understand how to add vectors. While traveling 10 km in the direction 60 degrees east of north the boat travels eastward a distance 10 cos(30) and northward 10 sin(30) - do you see why that is? Then add that east component to the original distance of 25 km east to get the total eastward movement of the boat. Now you can use Pythagoras to determine the distance the boat is away from its starting point (since you know the total north and east components of travel, which are two legs of a right triangle), and trigonometry (arc tangent of east divided by north) to determine the direction from the starting point in relation to north.