1985 AHSME Problems/Problem 13
Contents
Problem
Pegs are put in a board 
unit apart both horizontally and vertically. A rubber band is stretched over 
pegs as shown in the figure, forming a quadrilateral. Its area in square units is
![[asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));[/asy]](http://latex.artofproblemsolving.com/6/f/5/6f53bd3f40ff47916b824996e6c0b7cfa5d93c17.png)
Solution 1
![[asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7)); draw((0,0)--(4,0)--(4,3)--(0,3)--cycle); label("$A$",(0,3),NW); label("$B$",(4,3),NE); label("$C$",(4,0),SE); label("$D$",(0,0),SW); label("$E$",(1,3),N); label("$F$",(4,1),E); label("$G$",(3,0),S); label("$H$",(0,1),W); [/asy]](http://latex.artofproblemsolving.com/c/f/3/cf3df6ca185b3b9d326b6265dd0d26ae0a7ce655.png)
We draw in the rectangle bounding the given quadrilateral and label the points as shown. The area of rectangle 
is 
, while the areas of the triangles 
, 
, 
, and 
are, respectively, 
Hence the area of the given quadrilateral 
is 
.
Solution 2
The number of lattice points (i.e. pegs on the board) strictly inside the quadrilateral is 
and the number of lattice points on its boundary is . Therefore, by Pick's theorem, its area is 
.
See Also
| 1985 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
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