Final answer:
Yes, the triangle with vertices A(6,1), B(0,5), and C(-8,-7) is a right triangle.
Step-by-step explanation:
To determine if a triangle is a right triangle, we can use the Pythagorean theorem. According to the Pythagorean theorem, for a triangle to be a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) must be equal to the sum of the squares of the lengths of the other two sides.
Let's calculate the lengths of the sides of the given triangle ABC:
Side AB: √((0-6)^2 + (5-1)^2) = √((-6)^2 + 4^2) = √(36 + 16) = √52 = 2√13
Side BC: √((-8-0)^2 + (-7-5)^2) = √((-8)^2 + (-12)^2) = √(64 + 144) = √208 = 4√13
Side AC: √((-8-6)^2 + (-7-1)^2) = √((-14)^2 + (-8)^2) = √(196 + 64) = √260 = 2√65
Now, let's check if the triangle is a right triangle:
(2√13)^2 + (4√13)^2 = (2√65)^2
52 + 208 = 260
260 = 260
Since the two sides of the equation are equal, the triangle with vertices A(6,1), B(0,5), and C(-8,-7) is a right triangle.