Question

A golfer takes two putts to get the ball into the hole. The first putt rolls the ball 10.3 feet in the northwest direction and the second putt sends the ball due north 3.8 feet into the hole. How far was the ball originally from the hole?
11.23
12.41
13.26
Cannot be determined from the information given.

Answer 1

Let's put this on the usual Cartesian grid just so we can talk about it without drawing a picture.  We'll use map conventions, right is east, up is north.

The ball starts at (0,0).   10.3 feet northwest means we have an isosceles right triangle whose diagonal is 10.3 feet.  It's isosceles because northwest means equal parts north and west.   

The sides of these triangles are in ratio $1:1:\sqrt{2}$ so the coordinates after the first putt are

$(- 10.3 / \sqrt{2}, 10.3 / \sqrt{2})$

The negative sign indicates west, which doesn't really matter for this problem.  The distance from the origin to this point is 10.3 as required.

Now a second putt of 3.8 feet north puts us at
 
$(- 10.3 / \sqrt{2}, 10.3 / \sqrt{2} + 3.8)$

The squared distance to the origin is exactly

$d^2 = 10.3^2/2 + (10.3 / \sqrt{2} + 3.8)^2$

A little calculator work tells us

$d \approx 13.2621$

Third choice.