Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. Select the correct answer.
$$A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$$
A) $1,1 , \sqrt { 2 }$, isosceles
B) $1,1 , \sqrt { 2 }$, both
C) $3,2 , \sqrt { 5 }$, right
D) $1,2 , \sqrt { 5 }$, neither

Question

Quizplus · Accepted Answer

Using the distance formula, we have:
$AB = \sqrt{(1-(-1))^2 + (1-0)^2 + (-1-1)^2} = \sqrt{4+1+4} = 3$
$AC = \sqrt{(1-(-1))^2 + (1-0)^2 + (1-1)^2} = \sqrt{4+1} = \sqrt{5}$
$BC = \sqrt{(1-1)^2 + (1-1)^2 + (-1-1)^2} = \sqrt{4} = 2$

Since $AB 
eq AC 
eq BC$, the triangle is not isosceles. Also, since $AB^2 + BC^2 = 3^2 + 2^2 = 13$ and $AC^2 = (\sqrt{5})^2 = 5$, the triangle satisfies the Pythagorean theorem and is therefore a right triangle. Therefore, the answer is C.

Find the Length of Each Side of the Triangle ABCA B CABC

Find the Length of Each Side of the Triangle $A B C$