Problem:
In rectangle ABCD, we have A=(6,−22),B=(2006,178), and D=(8,y), for some integer y. What is the area of rectangle ABCD?
Answer Choices:
A. 4000
B. 4040
C. 4400
D. 40,000
E. 40,400
Solution:
The slope of line AB is (178−(−22))/(2006−6)=1/10. Since the line AD is perpendicular to the line AB, its slope is −10. This implies that
−10=8−6y−(−22), so y=−10(2)−22=−42, and D=(8,−42).
As a consequence,
AB=20002+2002=200101 and AD=22+202=2101
Thus
Area(ABCD)=AB⋅AD=400⋅101=(E)40,400
The problems on this page are the property of the MAA's American Mathematics Competitions