1. Using determinants, find the area of the triangle with vertices are `A(-3,5),B(3,-6),C(7,2)`

Solution:
The given points are `A(-3,5),B(3,-6),C(7,2)`

`:. x_1=-3,y_1=5,x_2=3,y_2=-6,x_3=7,y_3=2`

Area of a triangle `=1/2 |[x_1,y_1,1],[x_2,y_2,1],[x_2,y_2,1]|`

`=1/2 |[-3,5,1],[3,-6,1],[7,2,1]|`

`=1/2[-3 xx (-6 × 1 - 1 × 2) -5 xx (3 × 1 - 1 × 7) +1 xx (3 × 2 - (-6) × 7)]`

`=1/2[-3 xx (-6 -2) -5 xx (3 -7) +1 xx (6 +42)]`

`=1/2[-3 xx (-8) -5 xx (-4) +1 xx (48)]`

`=1/2[24 +20 +48]`

`=1/2[92]`

`=46`

Thus, the area of triangle is `46` square units

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2. Using determinants, find the area of the triangle with vertices are `A(-3,5),B(3,-6),C(7,2)`

Solution:
The given points are `A(-3,5),B(3,-6),C(7,2)`

`:. x_1=-3,y_1=5,x_2=3,y_2=-6,x_3=7,y_3=2`

Area of a triangle `=1/2 |[x_1,y_1,1],[x_2,y_2,1],[x_2,y_2,1]|`

`=1/2 |[-3,5,1],[3,-6,1],[7,2,1]|`

`=1/2[-3 xx (-6 × 1 - 1 × 2) -5 xx (3 × 1 - 1 × 7) +1 xx (3 × 2 - (-6) × 7)]`

`=1/2[-3 xx (-6 -2) -5 xx (3 -7) +1 xx (6 +42)]`

`=1/2[-3 xx (-8) -5 xx (-4) +1 xx (48)]`

`=1/2[24 +20 +48]`

`=1/2[92]`

`=46`

Thus, the area of triangle is `46` square units

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3. Using determinants, find the area of the triangle with vertices are `A(-2,-3),B(3,2),C(-1,-8)`

Solution:
The given points are `A(-2,-3),B(3,2),C(-1,-8)`

`:. x_1=-2,y_1=-3,x_2=3,y_2=2,x_3=-1,y_3=-8`

Area of a triangle `=1/2 |[x_1,y_1,1],[x_2,y_2,1],[x_2,y_2,1]|`

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

`=1/2[-2 xx (2 × 1 - 1 × (-8)) +3 xx (3 × 1 - 1 × (-1)) +1 xx (3 × (-8) - 2 × (-1))]`

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

`=1/2[-2 xx (10) +3 xx (4) +1 xx (-22)]`

`=1/2[-20 +12 -22]`

`=1/2[-30]`

`=-15`

`=15` (As area is positive)

Thus, the area of triangle is `15` square units

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4. Using determinants, find the area of the triangle with vertices are `A(1,0),B(6,0),C(4,3)`

Solution:
The given points are `A(1,0),B(6,0),C(4,3)`

`:. x_1=1,y_1=0,x_2=6,y_2=0,x_3=4,y_3=3`

Area of a triangle `=1/2 |[x_1,y_1,1],[x_2,y_2,1],[x_2,y_2,1]|`

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

`=1/2[1 xx (0 × 1 - 1 × 3) +0 xx (6 × 1 - 1 × 4) +1 xx (6 × 3 - 0 × 4)]`

`=1/2[1 xx (0 -3) +0 xx (6 -4) +1 xx (18 +0)]`

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

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

`=1/2[15]`

`=15/2`

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

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5. Using determinants, find the area of the triangle with vertices are `A(3,8),B(-4,2),C(5,1)`

Solution:
The given points are `A(3,8),B(-4,2),C(5,1)`

`:. x_1=3,y_1=8,x_2=-4,y_2=2,x_3=5,y_3=1`

Area of a triangle `=1/2 |[x_1,y_1,1],[x_2,y_2,1],[x_2,y_2,1]|`

`=1/2 |[3,8,1],[-4,2,1],[5,1,1]|`

`=1/2[3 xx (2 × 1 - 1 × 1) -8 xx (-4 × 1 - 1 × 5) +1 xx (-4 × 1 - 2 × 5)]`

`=1/2[3 xx (2 -1) -8 xx (-4 -5) +1 xx (-4 -10)]`

`=1/2[3 xx (1) -8 xx (-9) +1 xx (-14)]`

`=1/2[3 +72 -14]`

`=1/2[61]`

`=61/2`

Thus, the area of triangle is `61/2` square units

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6. Using determinants, find the area of the triangle with vertices are `A(-2,4),B(2,-6),C(5,4)`

Solution:
The given points are `A(-2,4),B(2,-6),C(5,4)`

`:. x_1=-2,y_1=4,x_2=2,y_2=-6,x_3=5,y_3=4`

Area of a triangle `=1/2 |[x_1,y_1,1],[x_2,y_2,1],[x_2,y_2,1]|`

`=1/2 |[-2,4,1],[2,-6,1],[5,4,1]|`

`=1/2[-2 xx (-6 × 1 - 1 × 4) -4 xx (2 × 1 - 1 × 5) +1 xx (2 × 4 - (-6) × 5)]`

`=1/2[-2 xx (-6 -4) -4 xx (2 -5) +1 xx (8 +30)]`

`=1/2[-2 xx (-10) -4 xx (-3) +1 xx (38)]`

`=1/2[20 +12 +38]`

`=1/2[70]`

`=35`

Thus, the area of triangle is `35` square units

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