If the points a, b and c have the coordinates a(5, 2), b(2, −3) and c(−8, 3), show that the triangle abc is a right-angled triangle.

A=5i+2j

B=2i-3j

C=-8i+3j

we find the vectors of AB, BC & AC:

AB=(2-5)i+(-3-2)j=-3i-5j

BC=(-8-2)i+(3-(-3))j=-10i+6j

AC=(-8-5)i+(3-2)j=-13i+j

now if the dot product of two of these vectors result in zero, it means that those vectors (representing the triangle sides) are perpendicular to each other (which means triangle ABC is a right-angled triangle)

let's try:

AB.BC=(-3)(-10)+(-5)(6)=30-30=0

thus the triangle ABC is a right-angled triangle.