If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
To prove this theorem, let's consider △ ABC with side lengths a, b, and c, where c is the longest side and c^2= a^2+ b^2.
Next, we will consider a line l and draw a segment DE on it such that its length is b.
At D, we draw a line m perpendicular to DE. On line m, we will locate a point F such that DF = a.
Now, we consider △ DEF. Since it is a right triangle, we can apply the Pythagorean Theorem and conclude that EF^2 = a^2 + b^2.
On the other hand, we have that c^2= a^2+ b^2 and so, we have the following relation.
FE^2 &= c^2
⇒
FE = c
From the above, we obtain that FE=AB. Let's list the congruent parts between △ ABC and △ EFD.
cc
BC ≅ DF & Side
AC ≅ DE & Side
AB ≅ EF & Side
In consequence, by the Side-Side-Side (SSS) Congruence Postulate, we conclude that △ ABC ≅ △ EFD. This implies that ∠ C ≅ ∠ D, so ∠ C is a right angle making △ ABC a right triangle.
