# 2005 AMC 8 Problems/Problem 7

## Problem

Bill walks $\tfrac12$ mile south, then $\tfrac34$ mile east, and finally $\tfrac12$ mile south. How many miles is he, in a direct line, from his starting point? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 1\tfrac14\qquad\textbf{(C)}\ 1\tfrac12\qquad\textbf{(D)}\ 1\tfrac34\qquad\textbf{(E)}\ 2$

## Solution

Draw a picture. $[asy] unitsize(3cm); draw((0,0)--(0,-.5)--(.75,-.5)--(.75,-1),linewidth(1pt)); label("\frac12",(0,0)--(0,-.5),W); label("\frac12",(.75,-.5)--(.75,-1),E); label("\frac34",(.75,-.5)--(0,-.5),N); draw((0,0)--(.75,0)--(.75,-1)--(0,-1)--cycle,gray); [/asy]$

Find the length of the diagonal of the rectangle to find the length of the direct line to the starting time using Pythagorean Theorem. $$\sqrt{\left(\frac12+\frac12\right)^2+\left(\frac34\right)^2} = \sqrt{1+\frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac54 = \boxed{\textbf{(B)}\ 1 \tfrac14}$$

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