The Converse of the Pythagorean Theorem

Rana is studying interior dungeon design at Fantasy U. With her dragon, Rekhi, by her side, Rana is building shelving brackets for her new dorm room. To be stable, the brackets must form RIGHT triangles. But Rekhi melted Rana's protractor, so she can't measure the angles. Nevertheless, Rana can work backwards and use the CONVERSE of the Pythagorean Theorem to determine if her brackets are RIGHT. Remember that the Pythagorean Theorem tells us that, IF the triangle is a right triangle and we call the legs 'a' and 'b' and the hypotenuse 'c' THEN 'a' squared plus 'b' squared equals 'c' squared. The CONVERSE of the Pythagorean Theorem reverses the order. IF 'a' squared plus 'b' squared equals 'c' squared THEN the triangle is a right triangle. In order to use the converse statement with confidence, we first need to PROVE that it's true. We'll use a technique called INDIRECT PROOF to do that. For any proof we always start with writing our statement we want to prove. To prove the converse of the pythagorean theorem means we're proving that if a triangle follows 'a' squared plus 'b' squared equals 'c' squared, then it is a right triangle. Next since we're doing an indirect proof, we assume the opposite of the statement we want to prove. Here let's ASSUME that our triangle is NOT a right triangle. We'll then use logic to show that this assumption leads to a CONTRADICTION. That contradiction will mean that our assumption was false. Therefore, the statement we wanted to prove must be true. We already have our statement that we want to prove: if a triangle follows 'a' squared plus 'b' squared equals 'c' squared, then it's a right triangle. We already know that we are going to assume that our triangle is NOT a right triangle. So what type of triangle could it be? It could be an ACUTE triangle or an OBTUSE triangle. So, for our proof let's draw an acute triangle... and an obtuse triangle. Let's label the vertices and the sides with 'A' 'B' and 'C' for both triangles. In the acute triangle, construct a perpendicular line segment to CB that is the same length as AC and call the new point 'D.' Therefore, this side length is also 'b'. We can construct the same perpendicular segment in the obtuse triangle as well. Now, we can complete our RIGHT triangles by connecting points 'D' and "B' in the acute triangle as well as in the obtuse triangle. Because these are right triangles the Pythagorean Theorem applies, so in our triangle 'a' squared plus 'b' squared equals 'DB' squared. But remember, we already know that 'a' squared plus 'b' squared equals 'c' squared. This means that we can call 'DB' 'c' in this triangle and in the other triangle. Now connect segment DA in both triangles, and recall that an isosceles triangle is a triangle that has two sides of equal length. Therefore focusing on the left diagram, we can see that we have TWO isosceles triangles here and here, right? Since ACD is an isosceles triangle, THESE two base angles are congruent. This means the measure of angle 'C', 'D', 'A' equals the measure of angle 'C', 'A', 'D'. And since triangle ABD is isosceles, THESE two base angles are congruent. This means the measure of angle 'B', 'D', 'A' equals the measure of angle 'B', 'A', 'D'. However, from the diagram, we can definitely see that angle C,D, A is GREATER than angle B,D,A. And because of THESE equations we can use the substitution property of equality, that means the measure of angle 'C', 'A', 'D' has to be greater than that of angle 'B', 'A', 'D'. But look back at the diagram: angle 'C', 'A', 'D' canNOT be greater than angle 'B', 'A', 'D', it's a CONTRADICTION! In fact, all those base angles didn't look congruent anyhow, but we had to prove they weren't congruent. Now let's look at the second diagram and using the same logic we get a similar situation because our final statement would say that angle 'C', 'D', 'A' is greater than angle 'B', 'D', 'A'. That's definitely a contradiction! Our ASSUMPTION, that our triangle WASN'T a right triangle, led to an overall CONTRADICTION, which was step four of our indirect proof. The only way for our triangle to possibly work is when it is a right triangle. Therefore, the CONVERSE of the Pythagorean Theorem is TRUE. IF 'a' squared plus 'b' squared equals 'c' squared, THEN the triangle is a right triangle. So, let's see if Rana's bracket is RIGHT. Here 'a' is 9 'b' is 12... and 'c' is 15. 9 squared is 81, 12 squared is 144, and 15 squared is 225. 81 plus 144 is 225, so 'a' squared plus 'b' squared DOES EQUAL 'c' squared. By the converse of the Pythagorean Theorem, that means the triangle is a right triangle. Rana can confidently construct a stable shelf using this bracket! Let's review! We proved the converse of the Pythagorean Theorem... using an INDIRECT PROOF. We started with the statement we wanted to prove. We assumed the converse was FALSE, and then used logic to lead to a CONTRADICTION. That meant the converse of the Pythagorean Theorem was actually TRUE. Let's see how Rana's shelf turned out! Best not to wake a sleeping dragon. That might lead to a serious contradiction!