The Pythagorean theorem is one of the most fundamental theorems in mathematics. It describes the relationship between the sides of a right triangle. In a right triangle with legs a and b, and hypotenuse c, the theorem states that a squared plus b squared equals c squared.
The Pythagorean theorem formula is a squared plus b squared equals c squared. Here, a and b represent the two legs of the right triangle, which are the sides that form the right angle. The letter c represents the hypotenuse, which is the longest side opposite the right angle.
Let's solve an example problem. We have a right triangle with legs of length 3 and 4. We need to find the length of the hypotenuse. We'll use the Pythagorean theorem to solve this.
Now let's solve step by step. First, we write the Pythagorean theorem formula. Then we substitute a equals 3 and b equals 4. This gives us 9 plus 16 equals c squared. Adding these together, we get 25 equals c squared. Taking the square root of both sides, we find that c equals 5. Therefore, the hypotenuse has length 5.
Let's verify our answer. We check that 3 squared plus 4 squared equals 5 squared. Indeed, 9 plus 16 equals 25, which is 5 squared. Our answer is correct! The hypotenuse of this right triangle is 5 units. This is a famous example known as the 3-4-5 triangle, one of the most common Pythagorean triples.