# Area of Heptagon

**Question**

The coordinates of seven vertices of a 7-gon are generated by the following rule: !!f(x)=(x-5)^2+3!! for !!x in(2,3,4,5,6,7,8)!!

Find the area of the polygon:

(a) 30

(b) 35

(c) 34

(d) 32

**Solution**

There are two ways to solve this question:

1) We can find the value of all vertices and apply "Shoelace Rule"

!!1/2 sum_(i=0)^(n-1) x_iy_(i+1)+x_ny_1- sum_(i=1)^(n-1)x_(i+1)y_i- x_1y_n!!

2)You can break the heptagon into two trapezium and one triangle.

In trapezium ABFG, a=6 , b=4 and h = 5. Therefore area is

!!A_1= (a+b)/2 xxh= 10/2 xx 5= 25!!

In trapezium BCFE, a=4, b=2 and h=3. Therefore area is

!!A_2= 6/2 xx3 = 9!!

Area of triangle CDE, !!A_3 = 1/2 xx 2 xx 1=1!!

So, area of heptagon will be

!!A= A_1+A_2+A_3= 25+9+1= 35!!

Therefore option (b) is correct.