What is the number of square units in the area of a triangle with vertices:?

Question

A (2,3) B (7,4) and C (4,6)?
Express your answer as a decimal to the nearest tenth.

Answer 1

1/2 |vectorAB XvectorAC| = 1/2| (5 ,1)x(2,3)|1/2(15-2) =6.5 units of area

Answer 2

Unless I have made a gross mistake, this is actually quite simple. When you plot out your triangle ABC on the x and y axis it touches in 3 places the sides of a rectangle which is 3 units X 5 units. The area of this rectangle is 15 square units.
The 3 points where the triangle touches this rectangle form 3 right angled triangles with the vertices of the rectangle. If we subtrcat the sum of the area of these 3 triangles from the rectangle, we get what is left over ie the area of triangle ABC.
The 3 triangles have a base of 5 units and vert. height of 1 unit, a base of 3 units and a vert height of 2 units, a base of 2 units and a vert height of 3 units, giving the areas of 5 square units, 3 square units and 3 square units respectively. Add these to make the area to be subtracted ie11 square units. Subtract 11 from 15 to get 4 square units, which is the area of the triangle ABC. Therefore your answer to the nearest tenth is 4.0 square units.

Answer 3

the formula is:
................... l x1 ... y1 ... 1 l
A = +/- 1/2 l x2 ... y2 ... 1l
....................l x3 ... y3 ... 1 l


................... l 2 ... 3 ... 1 l
A = +/- 1/2 l 7 ... 4 ... 1 l
....................l 4 ... 6 .... 1 l


from here you need to add another column of x and y

.................. l 2 ... 3 ... 1 l 2 ... 3
A = +/-1/2 l 7 ... 4 ... 1 l 7 ... 4
...................l 4 ... 6 .... 1 l 4 ... 6


A = +/-1/2 [ (2*4*1 + 3*1*4 + 1*7*6) - (4*4*1 + 6*1*2 + 1*7*3) ]

A = +/- 1/2 (62 - 49)

A = +/- 1/2 (13)

multiply to +1/2 to get a positive area
A = 6.5 unit^2