the vertices of a triangle are A(6.2, 7.4), B(1, 4.8) and C(6,2) find the area of the Triangle in km squared one unit on the graph represents 400 km.

Question

the vertices of a triangle are A(6.2, 7.4), B(1, 4.8) and C(6,2) find the area of the Triangle in km squared one unit on the graph represents 400 km.

Answer 1

AB = sqrt ((6.2 - 1)² + (7.4 - 4.8)² = 5.81

 

AC = sqrt ((6.2 - 6)² + (7.4 - 2)² = 5.40

 

BC = sqrt ((6 - 1)² + (2 - 4.8)² = 5.55

 

We see this is a scalene triangle, and so we can't use Pythagora's theorem to find the height as we would with a right-angled triangle. Using Hero's formula, s = 8.38.

 

So, the area = sqrt (8.38 * (8.38 - 5.81) * (8.38 - 5.4) * (8.38 - 5.55) ≈ 13.47 square units

 

Using the scale you provided, where one unit = 400 km (which means a square unit is 400*400 or 160,000 km² , the area = 13.47 * 160,000 km² = 2.1552 × 10⁶ km²

 

Answer 2

One method is to find the lengths of the three sides, using the distance formula.

Then use Hero's formula: area = √[s(s-a)(s-b)(s-c)]  where s is ½(a+b+c), the 'semiperimeter' and the side lengths are a,b,c.

 

There is also a method that uses a determinant, but you only encounter that method in an esoteric class.