Problem:
A piece of graph paper is folded once so that (0,2) is matched with (4,0), and (7,3) is matched with (m,n). Find m+n.
Answer Choices:
A. 6.7
B. 6.8
C. 6.9
D. 7.0
E. 8.0
Solution:
The crease in the paper is the perpendicular bisector of the segment that joins (0,2) to (4,0). Thus the crease contains the midpoint (2,1) and has slope 2, so the equation y=2x−3 describes it. The segment joining (7,3) and ( m,n ) must have slope −21, and its midpoint (27+m,23+n) must also satisfy the equation y=2x−3. It follows that
−21=m−7n−3 and 23+n=2⋅27+m−3, so 2n+m=13 and n−2m=5.
Solve these equations simultaneously to find that m=3/5 and n=31/5, so that m+n=34/5=6.8.
OR
As shown above, the crease is described by the equation y=2x−3. Therefore, the slope of the line through (m,n) and (7,3) is −1/2, so the points on the line can be described parametrically by (x,y)=(7−2t,3+t). The intersection of this line with the crease y=2x−3 is found by solving 3+t=2(7−2t)−3. This yields the parameter value t=8/5. Since t=8/5 determines the point on the crease, use t=2(8/5) to find the coordinates m=7−2(16/5)=3/5 and n=3+(16/5)=31/5.