Let's start by defining the three points as A(2,1), B(3,4), and C(-3,6). The first step to proving that these points form a right triangle is to calculate the lengths of the sides the points would form.
1. Calculate the distance between points A and B. This would gives us the length of side AB. We use the distance formula which is √[(x₂ - x₁)² + (y₂ - y₁)²]. Here, x₁ = 2, y₁ = 1, x₂ = 3, and y₂ = 4.
2. Similarly, calculate the distance between points B and C to get side BC.
3. Finally, calculate the distance between points C and A to get side AC.
Now that we have all three side lengths, we need to check if they satisfy the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is can be expressed as: a² + b² = c².
The side representing the hypotenuse is the longest side, so we order the three side lengths from shortest to longest. We can denote a and b as the lengths of the two shorter sides and c as the length of the longest side (the hypotenuse).
To check if the angles at point A, B, or C are right angles, we check if a² + b² equals c². If it does, then we have our right triangle! From our calculations, we found that this indeed is the case and therefore, the points A(2,1), B(3,4) and C(-3,6) do form a right triangle.