91Ó°ÊÓ

To find the lengths of the sides, we will use the distance formula, which in three dimensions is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). Let's find the length of each side: Side 1: Distance between \((a, 0, 0)\) and \((0, b, 0)\): \(AB = \sqrt{(0 - a)^2 + (b - 0)^2 + (0 - 0)^2} = \sqrt{a^2 + b^2}\) Side 2: Distance between \((0, b, 0)\) and \((0, 0, c)\): \(BC = \sqrt{(0 - 0)^2 + (0 - b)^2 + (c - 0)^2} = \sqrt{b^2 + c^2}\) Side 3: Distance between \((a, 0, 0)\) and \((0, 0, c)\): \(AC = \sqrt{(0 - a)^2 + (0 - 0)^2 + (c - 0)^2} = \sqrt{a^2 + c^2}\)

Now, we will use Heron's Formula to find the area of the triangle. First, calculate the semi-perimeter \(s = \frac{AB + BC + AC}{2}\): \(s = \frac{\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{a^2 + c^2}}{2}\) Now, we can use Heron's Formula to find the area: \(Area = \sqrt{s(s - AB)(s - BC)(s - AC)}\) Plugging in the values we found for the lengths of the sides: \(Area = \sqrt{\left(\frac{\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{a^2 + c^2}}{2}\right)\left(\frac{\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{a^2 + c^2}}{2} - \sqrt{a^2 + b^2}\right)\left(\frac{\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{a^2 + c^2}}{2} - \sqrt{b^2 + c^2}\right)\left(\frac{\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{a^2 + c^2}}{2} - \sqrt{a^2 + c^2}\right)}\) This formula represents the area of the triangle in terms of \(a, b,\) and \(c\).