Pythagorean theorem states that In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. In order for this to hold , the triangle be a right triangle. This is represented by the formula a[sup]2[/sup]+b[sup]2[/sup]=c[sup]2[/sup], where a and b are the sides adjacent to the right angle, and c is the hypotenuse.

In the below visual, We use the formula A[sup]2[/sup]+b[sup]2[/sup]=C[sup]2[/sup]. In our triangle, let a=the segment AB. b=BC, and c=AC. Toggle with the options to show the calculations and values of the sides to get a better understanding of the theorem.

Alter the side lengths by moving the point A and/or B to see how the values for A[sup]2 [/sup], B[sup]2 [/sup]and C[sup]2[/sup] change

Here we see when we look at triangle ABC, we could assume it's not a right triangle. However, when we evaluate a[sup]2[/sup]+b[sup]2[/sup]=c[sup]2[/sup], we see (3)[sup]2[/sup]+(4)[sup]2[/sup]=(5)[sup]2[/sup] [br]                9 + 16 = 25[br]                 25 =25 , here the triangle satisfies the pythagorean theorem, therefore triangle ABC IS a right triangle, although it may not appear like it. This is something your future teachers may try to fool you, remember that [b]if [/b][b]a[sup]2[/sup]+b[sup]2[/sup]=c[sup]2 [/sup]then the triangle is a right angle[/b]

We have seen that regardless of the arrangement of the points A,B,C, and the varying side lengths, that the pythagorean theorem always holds. This is a useful theorem to use moving forward this will be used in various levels of mathematical courses. We can also use this theorem to prove that triangles are right triangles, if a[sup]2[/sup]+b[sup]2[/sup]=c[sup]2[/sup], even if they're visualized incorrectly. If the triangle satisfies our formula, it is always a right triangle.