Problem:
What is the area enclosed by the geoboard quadrilateral below?
Answer Choices:
A. 15
B. 1821​
C. 2221​
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: 10×10=100.
Region A: 3×5=15.
Region B: 21​×6×7=21.
Region C: 21​×1×3=121​.
Region D: 21​×4×5=10.
Region E: 21​×6×10=30.
The area is 100−(15+21+121​+10+30)=100−7721​=2221​ 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 ​−1.
In this case, the area is 21+25​−1=2221​.
Answer: C​.
The problems on this page are the property of the MAA's American Mathematics Competitions
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