The Pythagorean theorem states that in a right triangle, the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. Does this mean that a triangle where 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared is necessarily a right triangle?
To answer this question, we’re gonna go through a list of questions to help us decide. So let us assume that triangle 𝐴𝐵𝐶 is of side lengths 𝑎, 𝑏, and 𝑐, with 𝑐 squared equal to 𝑎 squared plus 𝑏 squared. Let triangle 𝐷𝐵𝐶 be a right triangle of side lengths 𝑎, 𝑏, and 𝑑. Triangle 𝐴𝐵𝐶 is found here. And triangle 𝐷𝐵𝐶 is found here. So what do we know about each of these triangles? We know for triangle 𝐴𝐵𝐶, 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared. And we know for triangle 𝐷𝐵𝐶 that it’s a right triangle. So we essentially want to know, is triangle 𝐴𝐵𝐶 a right triangle?
So let’s begin with this first question. Using the Pythagorean theorem, what can you say about the relationship between 𝑎, 𝑏, and 𝑑?
So we are talking about triangle 𝐷𝐵𝐶. So we wanna know what can we say about the side lengths 𝑎, 𝑏, and 𝑑. Well, since it’s a right triangle, we can use the Pythagorean theorem. And it’s said to use that. And for the Pythagorean theorem, using the Pythagorean theorem, we know the square of the longest side is equal to the sum of the squares of the shorter sides. So which side is the longest side? That would be 𝑑, the hypotenuse. So 𝑑 squared is equal to the shorter sides squared. And then adding them together, so 𝑎 squared plus 𝑏 squared. So we could say 𝑑 squared is equal to 𝑎 squared plus 𝑏 squared.
Now our next question says, we know that for triangle 𝐴𝐵𝐶, 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared. What do you conclude about 𝑑?
So we are told that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared. But we know something else that’s equal to 𝑎 squared plus 𝑏 squared. That’s 𝑑 squared. This means 𝑐 squared and 𝑑 squared must be equal. And if 𝑐 squared is equal to 𝑑 squared, we could square root both sides and say that 𝑐 is equal to 𝑑.
The next question says, is it possible to construct different triangles with the same length sides?
So let’s take this triangle. If we were to take the pieces apart and reconstruct another triangle, will it look the same as the old one? It may be flipped upside down. But essentially, it’s still gonna be the same triangle. So is it possible to-to construct different triangles with the same length sides? The answer is no.
So what do you conclude about triangle 𝐴𝐵𝐶? Is it necessarily a right triangle? The fact that we know, 𝑐 is equal to 𝑑, we can replace 𝑐 with 𝑑. Then our side lengths would be exactly the same. And we concluded it’s not possible to construct different triangles with the same length sides. So if triangle 𝐷𝐵𝐶 is a right triangle, so is triangle 𝐴𝐵𝐶. Therefore, it is congruent to triangle 𝐷𝐵𝐶. So it is a right triangle at 𝑐.