AMC8 2005 8, Problem 7

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?

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. \[\sqrt{\left(\frac12+\frac12\right)^2+\left(\frac34\right)^2} = \sqrt{1+\frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac54 = \fbox{1\tfrac14}\]

Answer: 1\tfrac14