Okay, let's see. Bill is walking in different directions, and I need to figure out how far he is from his starting point in a straight line. Hmm, so he goes 1/2 mile south, then 3/4 mile east, and then another 1/2 mile south. Wait, so total southward distance is 1/2 + 1/2, which is 1 mile. And the eastward distance is 3/4 mile. So his path forms a right-angled triangle, right? The legs of the triangle would be the total south and east distances, and the hypotenuse would be the straight-line distance back to the start. So I can use the Pythagorean theorem here.

But let me make sure I'm not making a mistake. Let me visualize his path. He starts at a point, walks south half a mile, then turns east and walks three-quarters of a mile, then turns south again and walks another half mile. So his final position is 1 mile south and 3/4 mile east from the starting point. Yeah, that seems right. So the straight-line distance would be the square root of (1^2 + (3/4)^2). Let me compute that.

First, 1 squared is 1. Then, 3/4 squared is 9/16. Adding those together gives 1 + 9/16. To add them, I need a common denominator. 1 is 16/16, so 16/16 + 9/16 is 25/16. The square root of 25/16 is 5/4, which is 1.25 miles. Wait, but 5/4 is 1.25, which is already to the nearest hundredth. So the answer should be 1.25 miles. But let me double-check my steps.

Wait, total south is 1/2 + 1/2 = 1 mile. East is 3/4 mile. So the legs are 1 and 3/4. Squaring each: 1^2 = 1, (3/4)^2 = 9/16. Sum is 25/16. Square root of 25/16 is 5/4. 5 divided by 4 is indeed 1.25. So that seems correct. But let me confirm if there's any other component I might have missed.

Wait, the problem says he walks 1/2 mile south, then 3/4 mile east, then 1/2 mile south. So that's two south segments and one east. So total south is 1/2 + 1/2 = 1. Correct. So the coordinates would be (east, south). Starting at (0,0), after first south walk, (0, -1/2). Then east to (3/4, -1/2). Then south again to (3/4, -1). So final position is (3/4, -1). The distance from origin is sqrt( (3/4)^2 + (1)^2 ) = sqrt(9/16 + 16/16) = sqrt(25/16) = 5/4 = 1.25. So yes, that's right. So the answer is 1.25 miles. To the nearest hundredth, it's already exact at 1.25, so no rounding needed. Therefore, the answer is 1.25.
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