Answer:
The area of the triangle is 18 square units.
Step-by-step explanation:
First, we determine the lengths of segments AB, BC and AC by Pythagorean Theorem:
AB
[tex]AB = \sqrt{(5-2)^{2}+[6-(-1)]^{2}}[/tex]
[tex]AB \approx 7.616[/tex]
BC
[tex]BC = \sqrt{(-1-5)^{2}+(4-6)^{2}}[/tex]
[tex]BC \approx 6.325[/tex]
AC
[tex]AC = \sqrt{(-1-2)^{2}+[4-(-1)]^{2}}[/tex]
[tex]AC \approx 5.831[/tex]
Now we determine the area of the triangle by Heron's formula:
[tex]A = \sqrt{s\cdot (s-AB)\cdot (s-BC)\cdot (s-AC)}[/tex] (1)
[tex]s = \frac{AB+BC + AC}{2}[/tex] (2)
Where:
[tex]A[/tex] - Area of the triangle.
[tex]s[/tex] - Semiparameter.
If we know that [tex]AB \approx 7.616[/tex], [tex]BC \approx 6.325[/tex] and [tex]AC \approx 5.831[/tex], then the area of the triangle is:
[tex]s \approx 9.886[/tex]
[tex]A = 18[/tex]
The area of the triangle is 18 square units.
