<Question>
Bill walks $\frac{1}{2}$ mile south, then $\frac{3}{4}$ mile east, and finally $\frac{1}{2}$ mile south. How many miles is he, in a direct line, from his starting point?  Express your answer as a decimal to the nearest hundredth.
</Question>

<think>
<Plan_1>
Okay, let's see. Bill is walking in different directions, and we need to find out how far he is from his starting point in a straight line. Hmm, directions are south, then east, then south again. So, maybe I should break this down into parts and visualize his path.

<prompt_1.1> Visualize Bill's path by breaking down his movements into south and east components. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, he walks 1/2 mile south. Then 3/4 mile east, and then another 1/2 mile south. So, total southward movement is 1/2 + 1/2, which is 1 mile south. Then, he went 3/4 mile east. So, his final position is 1 mile south and 3/4 mile east from where he started.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
To find the straight-line distance from the starting point, this forms a right triangle where one leg is the total south distance and the other leg is the east distance.

<prompt_2.1> Calculate the straight-line distance using the Pythagorean theorem based on the above plan. </prompt_2.1>

But wait, let me double-check. Let's map out each step.

<prompt_2.2> Map out each step of Bill's walk to verify the total distance calculation based on the above plan. </prompt_2.2>

Wait a second, maybe I should check if there's any trick here. Like, does the order matter?

<prompt_2.3> Consider if the order of Bill's steps affects the final position and calculation based on the above plan. </prompt_2.3>

Let me verify using the Pythagorean theorem. The legs of the triangle are 1 mile and 0.75 miles.

<prompt_2.4> Verify the straight-line distance using the Pythagorean theorem based on the above plan. </prompt_2.4>

But wait, maybe I made a mistake in the order. Let me visualize again. Starting at the origin. First, walks 1/2 mile south.
If south is negative y-direction, that would be (0, -0.5). Then 3/4 mile east to (0.75, -0.5). Then another 1/2 mile south to (0.75, -1.0). So, the coordinates are (0.75, -1.0). The displacement from (0,0) is sqrt(0.75^2 + 1.0^2).
</Plan_2>

<Execution_2>
<execution_2.1>
The Pythagorean theorem says that the distance (let's call it 'd') is the square root of (south distance squared plus east distance squared). So, d = sqrt((1 mile)^2 + (0.75 mile)^2). Let me compute that. 

First, 1 squared is 1, and 0.75 squared is 0.5625. Adding those together gives 1 + 0.5625 = 1.5625. Then the square root of 1.5625. Hmm, what's the square root of 1.5625? Let me think. I know that 1.25 squared is 1.5625 because 1.25 * 1.25 = 1.5625. So, the distance is 1.25 miles.
</execution_2.1>

<execution_2.2>
First step: 1/2 mile south. So, he's half a mile down. Then, he turns east and walks 3/4 mile. So, from that half-mile south point, he moves east. Then, again, half a mile south from there. So, total south is 1/2 + 1/2 = 1 mile. East is just 3/4 mile. So, his displacement should be 1 mile south and 3/4 mile east.
</execution_2.2>

<execution_2.3>
Because he first goes south, then east, then south again. So, the total southward movement is indeed 1/2 + 1/2 = 1 mile. The eastward is only 3/4 mile. So, the displacement should be the hypotenuse of a right triangle with sides 1 mile and 0.75 miles.
</execution_2.3>

<execution_2.4>
So, the distance is sqrt(1^2 + 0.75^2). Let's calculate that. 1 squared is 1, 0.75 squared is 0.5625. Adding them gives 1 + 0.5625 = 1.5625. The square root of 1.5625 is...
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.4:
Let me check. 1.25 times 1.25 is 1.5625.
Yes, because 1.2 * 1.2 = 1.44, 1.25 * 1.25 = 1.5625. So, sqrt(1.5625) = 1.25. Therefore, the straight-line distance is 1.25 miles.

But the question says to express the answer as a decimal to the nearest hundredth.
Well, 1.25 is already to the nearest hundredth. Because 1.25 is the same as 1.250000...

<prompt_3.1> Round the result to the nearest hundredth based on the above plan. </prompt_3.1>

Hmm, seems straightforward. But let me just make sure there's no detail I missed.

<prompt_3.2> Confirm the solution and ensure no details were missed based on the above plan. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
, so rounding to two decimal places, it's 1.25. So, the answer is 1.25 miles.

Wait, but let me confirm once more. Maybe I made a mistake in adding the fractions? Let's see. The south legs are 1/2 and 1/2. 1/2 is 0.5, so 0.5 + 0.5 = 1.0. The east leg is 3/4, which is 0.75. So, the legs are 1.0 and 0.75. Then, Pythagoras gives sqrt(1 + 0.5625) = sqrt(1.5625) = 1.25. Yep, that's correct.
</execution_3.1>

<execution_3.2>
The problem says "in a direct line, from his starting point". So, straight line distance, which is the displacement. Since he moves in south, east, south, the final position is 1 mile south and 0.75 miles east. So, the displacement is calculated as the hypotenuse. So, yes, 1.25 miles.

Wait, but maybe the directions are different? Like, south is 180 degrees? Wait, no, in standard terms, south is down, east is right. So, plotting the points, the displacement would be the straight line from start to end. So, yes, the calculation is correct.

Alternatively, maybe using coordinate system. Let's assign coordinates. Let's say starting point is (0, 0). Moving south 1/2 mile would take him to (0, -0.5). Then moving east 3/4 mile would take him to (0.75, -0.5). Then moving south another 1/2 mile would take him to (0.75, -1.0). So, final coordinates (0.75, -1.0). The distance from (0,0) is sqrt((0.75)^2 + (-1.0)^2) = sqrt(0.5625 + 1.0) = sqrt(1.5625) = 1.25. So, yes, 1.25 miles. So, 1.25 is already to the nearest hundredth. So, the answer is 1.25.

I think that's correct. I don't see any errors in the reasoning.
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, the final answer, rounded to the nearest hundredth, is $\boxed{1.25}$ miles.
</Final_answer>
</think>