Correct option is (b) scalene; \( \frac{-3+6\sqrt 3 }2\) sq. units
Distance between two points = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Distance between the points A(3, 1) and B(−3, 2) = \(\sqrt{(-3-3)^2 + (2 - 1)^2}\)
\(= \sqrt{36 + 1}\)
\(= \sqrt{37}\)
Distance between the points B(−3, 2) and C(0, 2 − √3) = \(\sqrt{(0+3)^2 + (2 - \sqrt 3 -2)^2}\)
\(=\sqrt {9 + 3}\)
\(= \sqrt {12}\)
Distance between the points A(3, 1) and C(0, 2−√3)
\(\sqrt{(0-3)^2 + (2 - \sqrt 3 -1)^2}\)
\(=\sqrt {9 +1+ 3-2\sqrt 3}\)
\(= \sqrt {13 - 2\sqrt 3}\)
Since the length of the sides between all vertices are different, they are the vertices of a scalene triangle.
Area of △ABC = \(\left|\frac{x_1(y_2 -y_3)+x_2(y_3 - y_1) + x_3(y_1 - y_2)}{2}\right|\)
\(= \left|\frac{3(2 - 2 + \sqrt 3)+ (-3)(2- \sqrt 3 -1) + 0(1 - 2)}{2}\right|\)
\(= \left|\frac{3\sqrt 3 -3+ 3\sqrt 3}2\right|\)
\(= \frac{-3+6\sqrt 3 }2\)
