Answer: To find the area of a triangle, we need to calculate the length of its sides or use the cross product of two of its sides. Let's start by finding the length of each side of the triangle.
The first side connects points \( (1,-1,3) \) and \( (-2,-5,4) \). Using the distance formula:
\[ \text{Side 1} = \sqrt{(-2 - 1)^2 + (-5 - (-1))^2 + (4 - 3)^2} \]
\[ = \sqrt{ (-3)^2 + (-4)^2 + 1^2 } \]
\[ = \sqrt{9 + 16 + 1} \]
\[ = \sqrt{26} \]
The second side connects points \( (-2,-5,4) \) and \( (3,1,-4) \). Again, using the distance formula:
\[ \text{Side 2} = \sqrt{(3 - (-2))^2 + (1 - (-5))^2 + (-4 - 4)^2} \]
\[ = \sqrt{ (5)^2 + (6)^2 + (-8)^2 } \]
\[ = \sqrt{25 + 36 + 64} \]
\[ = \sqrt{125} \]
\[ = 5\sqrt{5} \]
The third side connects points \( (3,1,-4) \) and \( (1,-1,3) \). Using the distance formula again:
\[ \text{Side 3} = \sqrt{(1 - 3)^2 + (-1 - 1)^2 + (3 - (-4))^2} \]
\[ = \sqrt{ (-2)^2 + (-2)^2 + (7)^2 } \]
\[ = \sqrt{ 4 + 4 + 49 } \]
\[ = \sqrt{57} \]
Now that we have the lengths of all three sides, we can use Heron's formula to find the area. Heron's formula states:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a, b, \) and \( c \) are the lengths of the sides of the triangle, and \( s \) is the semi-perimeter of the triangle, calculated as:
\[ s = \frac{a+b+c}{2} \]
Let's substitute the values into the formula:
\[ s = \frac{\sqrt{26} + 5\sqrt{5} + \sqrt{57}}{2} \]
Now calculate the area using Heron's formula:
\[ \text{Area} = \sqrt{\frac{\sqrt{26} + 5\sqrt{5} + \sqrt{57}}{2}\left(\frac{\sqrt{26} + 5\sqrt{5} + \sqrt{57}}{2} - \sqrt{26}\right)\left(\frac{\sqrt{26} + 5\sqrt{5} + \sqrt{57}}{2} - 5\sqrt{5}\right)\left(\frac{\sqrt{26} + 5\sqrt{5} + \sqrt{57}}{2} - \sqrt{57}\right)} \]
Calculating this expression might be messy, so let's simplify the formula first. We can simplify it by multiplying both sides of the equation by 4 and then squaring both sides. This will give us:
\[ \text{Area}^2 = (\sqrt{26} + 5\sqrt{5} + \sqrt{57})\left((\sqrt{26} + 5\sqrt{5} + \sqrt{57}) - 2\sqrt{26}\right)\left((\sqrt{26} + 5\sqrt{5} + \sqrt{57}) - 10\sqrt{5}\right)\left((\sqrt{26} + 5\sqrt{5} + \sqrt{57}) - 2\sqrt{57}\right) \]
\[ \text{Area}^2 = (\sqrt{26} + 5\sqrt{5} + \sqrt{57})(7\sqrt{26} - \sqrt{26} + 5\sqrt{5})(-3\sqrt{5} + \sqrt{26} + 5\sqrt{5})(5\sqrt{5} + \sqrt{26} - \sqrt{57}) \]
\[ \text{Area}^2 = (\sqrt{26} + 5\sqrt{5} + \sqrt{57})(6\sqrt{26} + 10\sqrt{5})(2\sqrt{5} + \sqrt{26})(19\sqrt{5} + 5\sqrt{26} - 3\sqrt{57}) \]
Calculating the expression is still challenging, so we can use a calculator or an algebraic software to evaluate it.
The final answer will be the square root of the calculated value