The vertices of a triangle are given. Use the distance formula to find what kind of triangle each is. A sketch of the graph might be helpful. Note: An isosceles triangle has two equal sides. An equilateral triangle has three equal sides. A right triangle satisfies the Pythagorean Theorem.
(6, 7), (4, 4), (6, 1)
A) equilateral isosceles right
B) isosceles
C) right, not isosceles
D) equilateral
E) equilateral right

Question

Quizplus · Accepted Answer

Using the distance formula, we find that the side lengths of the triangle are:
AB = $\sqrt{(4-6)^2 + (4-7)^2}$ = $\sqrt{10}$
AC = $\sqrt{(6-6)^2 + (1-7)^2}$ = 6
BC = $\sqrt{(4-6)^2 + (4-1)^2}$ = $\sqrt{13}$

Since two sides have different lengths, the triangle is not equilateral. Also, none of the sides have the same length, so the triangle is not isosceles.

To check for a right triangle, we need to see if any of the side lengths satisfy the Pythagorean Theorem. However, $\sqrt{10}^2$ + $\sqrt{13}^2$ ≠ 6$^2$, so the triangle is not right. Therefore, the triangle must be scalene.

Thus, the best choice is B) isosceles.

The Vertices of a Triangle Are Given