Given points A (2; 4), B (6; -4) and C (-8; -1) Prove that triangle ABC is right-angled with hypotenuse BC.

Let’s apply the formula for determining the length of a line segment by its coordinates.

D = √ ((X2 – X1) ^ 2 + (Y2 – Y1) ^ 2).

Determine the length of the segment AB.

AB = √ (6 – 2) ^ 2 + (-4 – 4) ^ 2 = √ (16 + 64) = √80 cm.

Determine the length of the segment AC.

AC = √ (-8 – 2) ^ 2 + (-1 – 4) ^ 2 = √ (100 + 25) = √125.

Determine the length of the segment BC.

ВС = √ (-8 – 6) ^ 2 + (-1 – (-4)) ^ 2 = √ (196 + 9) = √205.

Let’s check the Pythagorean theorem:

BC ^ 2 = 205.

AC ^ 2 + AB ^ 2 = 125 + 80 = 205.

BC ^ 2 = AC ^ 2 + BC ^ 2, hence the triangle ABC is rectangular, and BC is its hypotenuse, which was required to prove.