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. $$ (2,-7,3),(-1,5,8),(4,6,-1) $$

Short answer

Performing the above steps will allow the determination of whether the triangle is acute, obtuse, or right-angled.

Step-by-step solution

Compute the distace of three sides

Compute the lengths of the sides (d1, d2, d3), using the distance formula. \[ d=\sqrt{(x_2-x_1)^{2} +(y_2-y_1)^{2}+(z_2-z_1)^{2}} \] where (x1, y1, z1) and (x2, y2, z2) are the coordinates of two points in the space.

Compute the distance of sides

Perform the calculations to find the lengths of the sides. Consider the given points as A(2,-7,3), B(-1,5,8), and C(4,6,-1), to compute the distances as: AB (d1), BC (d2), and AC (d3).

Compute squares of three sides

Now, square each of the lengths obtained above (d1^2, d2^2, and d3^2).

Determine the type of the Triangle

Depending on the values of d1^2, d2^2, d3^2, determine whether the triangle is right, acute, or obtuse. If any of these values is equal to the sum of the other two, then it is a right-angled triangle. If none of them is equal to the sum of the other two, compare them: If any one is greater than the sum of the other two, then the triangle is obtuse. Otherwise, if each one is less than the sum of the other two, then the triangle is acute.