Problem:
What is the area enclosed by the geoboard quadrilateral below?
Answer Choices:
A. 15
B. 1821β
C. 2221β
D. 27
E. 41
Solution:
To find the area, subtract the areas of regions A,B,C,D and E from that of the surrounding square.
Square: 10Γ10=100.
Region A: 3Γ5=15.
Region B: 21βΓ6Γ7=21.
Region C: 21βΓ1Γ3=121β.
Region D: 21βΓ4Γ5=10.
Region E: 21βΓ6Γ10=30.
The area is 100β(15+21+121β+10+30)=100β7721β=2221β square units.
OR
By Pick's Theorem, the area of a polygon with vertices in a lattice is
(number of points inside) +2 number of points on boundary ββ1.
In this case, the area is 21+25ββ1=2221β.
Answer: Cβ.
The problems on this page are the property of the MAA's American Mathematics Competitions