Final answer:
To find the area of the triangle with given vertices, calculate the length of the sides using the distance formula and use Heron's formula to find the area.
Step-by-step explanation:
To find the area of a triangle with given vertices, we can use the formula for the area of a triangle using coordinates. Let's label the given vertices as A(2,4,-3), B(1,-3,2), and C(-1,-4,-2). We can calculate the length of the three sides of the triangle using the distance formula, and then use Heron's formula to find the area.
Using the distance formula, we find that AB = sqrt((1-2)^2 + (-3-4)^2 + (2-(-3))^2) = sqrt(95), BC = sqrt((1-(-1))^2 + (-3-(-4))^2 + (2-(-2))^2) = sqrt(32), and AC = sqrt((2-(-1))^2 + (4-(-4))^2 + (-3-(-2))^2) = sqrt(54).
Now, we can use Heron's formula to find the area. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by A = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter of the triangle. In this case, s = (AB+BC+AC)/2 = (sqrt(95)+sqrt(32)+sqrt(54))/2. Plugging in the values, we calculate A = sqrt(s(s-a)(s-b)(s-c)) = sqrt(((sqrt(95)+sqrt(32)+sqrt(54))/2)(((sqrt(95)+sqrt(32)+sqrt(54))/2)-sqrt(95)))(((sqrt(95)+sqrt(32)+sqrt(54))/2)-sqrt(32))(((sqrt(95)+sqrt(32)+sqrt(54))/2)-sqrt(54))). This gives us the area of the triangle.