Problem:
The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment AB meets segment CD at E. Find the length of segment AE.
Answer Choices:
A. 45​/3
B. 55​/3
C. 125​/7
D. 25​
E. 565​/9
Solution:
Extend DC to F. Triangle FAE and DBE are similar with ratio 5:4. Thus AE=5⋅AB/9,AB=32+62​=45​=35​, and AE=5(35​)/9=55​/3.
OR
Coordinatize the points so that A=(0,3),B=(6,0),C=(4,2), and D=(2,0). Then the line through A and B is given by x+2y=6, and the line through C and D is given by x−y=2. Solve these simultaneously to get E=(310​,34​). Hence AE=(310​−0)2+(34​−3)2​=9125​​=355​​.
