Given the coordinates of the vertices of the triangle MPT: M (-4; 3), P (2; 7), T (8; -2) Prove that the given triangle is right-angled

Given the coordinates of the vertices of the triangle MPT: M (-4; 3), P (2; 7), T (8; -2) Prove that the given triangle is right-angled and find the radius of the circumscribed circle around it.

M (-4; 3), P (2; 7), T (8; -2)

Find the coordinates of the following vectors:

MP = (2 – (-4); 7 – 3) = (6; 3);

PT = (8 – 2; -2 – (-7)) = (6; 5);

TM = (-4 – 8; 3 – (-2)) = (-12; 5).

Let’s calculate their modules:

| MP | = √ (6 ^ 2 + 3 ^ 2) = √45;

| PT | = √ (6 ^ 2 + 5 ^ 2) = √61;

| TM | = √ (-12 ^ 2 + 5 ^ 4) = √106.

Then since | MP | ^ 2 + | PT | ^ 2 = | TM | ^ 2, the Pythagorean theorem holds. Hence the triangle is rectangular.

The radius R of the circumscribed circle about a right-angled triangle is half the hypotenuse:

R = √106 / 2 = √ (53/2).