For this case we have to find the area through Heron's formula:
[tex]A = \sqrt{s (s-a) (s-b) (s-c)}[/tex]
Where:
s: It's half the perimeter of the triangle
a, b, c are the sides.
We can find the sides by equating the distance between two points:
[tex]S: (- 2, -2)\\Q: (- 1,2)\\R: (1, -4)[/tex]
[tex]SQ=\sqrt {(x2-x1) ^ 2+(y2-y1) ^ 2}\\SQ=\sqrt {(- 1 + 2) ^ 2+ (2 + 2) ^ 2}\\SQ=\sqrt {1 ^ 2 +(4) ^2}\\SQ=\sqrt {17}\\SQ=4.12[/tex]
We found QR:
[tex]QR = \sqrt {(x2-x1) ^ 2 + (y2-y1) ^ 2}\\QR = \sqrt {(1 + 1) ^ 2 + (- 4-2) ^ 2}\\QR = \sqrt {(2) ^ 2 + (- 6) ^ 2}\\QR = \sqrt {40}\\QR = 6.33[/tex]
We found RS:
[tex]RS=\sqrt{(x2-x1)^2+(y2-y1)^2}\\RS=\sqrt{(1 + 2)^2+(- 4 + 2)^2}\\RS=\sqrt{(3)^2+(- 2)^2}\\RS=\sqrt{9+4}\\RS=\sqrt {13}\\RS=3.60[/tex]
So, half of the perimeter is:
[tex]s = \frac {4.12 + 6.33 + 3.61} {2} = 7.03[/tex]
Thus, the area is:
[tex]A = \sqrt{7.03 (7.03-4.12) (7.03-6.33) (7.03-3.61)}\\A = \sqrt{7.03 (2.91) (0.7) (3.42)}\\A = \sqrt {48.97}\\A = 6,997[/tex]
Rounding we have that the area is 7 square units
Answer:
Option A
