Points A (-10; 3), B (2; 9), C (3; 7) are given. Prove that triangle ABC is right-angled.

Let’s find the coordinates of the vector AB. AB (2 – (- 10); 9-3), that is, AC (12; 6).
Let’s find the coordinates of the vectors AC and BC:
AC (3 – (- 10), 7-3), that is, AC (13.4)
ВС (3-2,7-9), i.e. ВС (1, -2)
Let’s find the lengths of the vectors AB, BC, AC.
| AB | = √ (12 ^ 2 + 6 ^ 2) = √ (144 + 36) = √180
| ВС | = √ (1 ^ 2 + (- 2) ^ 2) = √ (1 + 4) = √ 5
| AC | = √ (13 ^ 2 + 4 ^ 2) = √ (169 + 16) = √185
Since AB ^ 2 + BC ^ 2 = 180 + 5 = 185 = AC ^ 2, then by the theorem converse to the Pythagorean theorem, the triangle ABC is right-angled.