Question

Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these. $$ (5,3,4),(7,1,3),(3,5,3) $$

Short answer

The lengths of the sides of the triangle are 3, 3, and 4. The triangle is both a right triangle and an isosceles triangle.

Step-by-step solution

Calculate the Length of Each Side

Use the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\) for each pair of points provided. Hence, we get the lengths of AB, BC, and AC as follows:\nAB = \(\sqrt{(7-5)^2 + (1-3)^2 + (3-4)^2} = \sqrt{4+4+1}=3\),\n BC = \(\sqrt{(3-7)^2 + (5-1)^2 + (3-3)^2} = \sqrt{16+16+0}=4\)\nand AC = \(\sqrt{(5-3)^2 + (3-5)^2 + (4-3)^2} = \sqrt{4+4+1}=3\).

Check the Type of Triangle

We see that AB = AC, so the triangle is isosceles. Also, since we have \(AB^2 + AC^2 = BC^2\), which is \(3^2 + 3^2 = 4^2\), thus the triangle is also a right triangle.