Question

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-3,4),(0,1,2),(-1,2,0) $$

Short answer

Perform the calculations and verify using the Pythagorean theorem as described in steps one through three to determine whether the triangle is right-angled, acute, or obtuse.

Step-by-step solution

Calculate the lengths of the sides

Firstly, apply the distance formula to calculate the length of each side of the triangle. The formula is given by: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\] Apply this to determine the distances between each set of points, which will be the lengths of the sides of the triangle.

Sort the lengths

Arrange the lengths of the sides in increasing order. The largest length (hypotenuse in case of right-angled triangle) should be at the last. This arrangement will assist in applying the Pythagorean theorem properly.

Apply the Pythagorean theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, check if \[(length_1)^2 + (length_2)^2 = (length_3)^2\] If this equation holds true, the triangle is right-angled. If \[(length_1)^2 + (length_2)^2 > (length_3)^2\] the triangle is acute. If \[(length_1)^2 + (length_2)^2 < (length_3)^2\] the triangle is obtuse.