Find area of the triangle with vertices at the point given in each of the following :
(i) `(1, 0),(6, 0), (4, 3)`
(ii) `(2, 7), (1, 1),(10 , 8) `
(iii) `(-2, -3), (3, 2), (-1, -8)`

Explanation

(i) `(1, 0),(6, 0), (4, 3)`

The area of triangle is given by,

`Delta=1/2|[x_1,y_1,1],[x_2,y_2,1],[x_3,y_3,1]|`

Here,
` " " x_1=1, " " " y_1=0`

` " " x_2=6, " " " y_2=0`

` " " x_3=4, " " " y_3=3`

`Delta=1/2|[1,0,1],[6,0,1],[4,3,1]|`

` " " " =1/2(1|[0,1],[3,1]|-0|[6,1],[4,1]|+1|[6,0],[4,3]|)`

` " " " =1/2(1(0-3)-0(6-4)+1(18-0))`

` " " " =1/2(1(-3)+0+1(18))=1/2[-3+18]`

`Delta=15/2`

Thus,
the required area of triangle is `15/2` square units.


(ii) `(2, 7), (1, 1),(10 , 8) `

The area of triangle is given by,

`Delta=1/2|[x_1,y_1,1],[x_2,y_2,1],[x_3,y_3,1]|`

Here,
` " " x_1=2, " " " y_1=7`

` " " x_2=1, " " " y_2=1`

` " " x_3=10, " " " y_3=8`

`Delta=1/2|[2,7,1],[1,1,1],[10,8,1]|`

` " " " =1/2(2|[1,1],[8,1]|-7|[1,1],[10,1]|+1|[1,1],[10,8]|)`

` " " " =1/2(2(1-8)-7(1-10)+1(8-10))`

` " " " =1/2(2(-7)-7(-9)+1(-2))`

` " " " =1/2(-14+63-2)=1/2(-16+63)`

`Delta=47/2`

Thus,
the required area of triangle is `47/2` square units.


(iii) `(-2, -3), (3, 2), (-1, -8)`

The area of triangle is given by,

`Delta=1/2|[x_1,y_1,1],[x_2,y_2,1],[x_3,y_3,1]|`

Here,
` " " x_1=-2, " " " y_1=-3`

` " " x_2=3, " " " y_2=2`

` " " x_3=-1, " " " y_3=-8`

`Delta=1/2|[-2,-3,1],[3,2,1],[-1,-8,1]|`

` " " " =1/2(-2|[2,1],[-8,1]|-(-3)|[3,1],[-1,1]|+1|[3,2],[-1,-8]|)`

` " " " =1/2[-2(2-(-8))+3(3-(-1))+1(-24-(-2)]`

` " " " =1/2[-2(2+8)+3(3+1)+1(-24+2)]`

` " " " =1/2[-2(10)+3(4)+1(-22)]=1/2[-20-22+12]=1/2(-30)`

`Delta=-15`

Since the area of triangle is always positive,

Thus,
he required area of triangle is `15` square units.