Answer
(a)
$AC^2=AB^2+BC^2$
$50=10+40$
$50=50$
(b) $Area=10$
Work Step by Step
(a) According to the definition of Pythagorean Theorem, square of the biggest side of a triangle equals to the sum of squares of the other sides.
Let's first calculate length of each side using distance formula:
$AB=\sqrt{(3-2)^2+(-1-2)^2}=\sqrt{1+9}=\sqrt{10}$
$BC=\sqrt{(-3-3)^2+(-3+1)^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$
$AC=\sqrt{(-3-2)^2+(-3-2)^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}$
As we see, $AC$ is the biggest one, so to show whether the triangle is right or not we can write:
$AC^2=AB^2+BC^2$
$50=10+40$
$50=50$
(b) Area of triangle is $Height \times Base \times \frac{1}{2}$. In case of right triangle, Height and Base are $adjacent$ and $opposite$:
$A=\sqrt{10}\times 2\sqrt{10} \times \frac{1}{2}=10$