How does the length of the hypotenuse in a right triangle compare to the lengths of the legs?
In a right triangle, the length of the hypotenuse is always longer than the lengths of its legs. This is a direct result of the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the two legs. Mathematically, the theorem is expressed as:
c² = a² + b²
Where:
- c represents the length of the hypotenuse
- a and b represent the lengths of the legs
Since the lengths of a and b are squared and then added together to determine the length of the hypotenuse, it is evident that the hypotenuse will always be longer than either of the legs.
In a right triangle, the length of the hypotenuse can be determined using the Pythagorean theorem. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the two legs.
If the lengths of the legs are represented by 'a' and 'b', and the length of the hypotenuse is represented by 'c', then the Pythagorean theorem can be expressed as:
c^2 = a^2 + b^2
From this equation, we can determine that the length of the hypotenuse is always greater than the lengths of the legs. This is because the squares of the leg lengths are added together and then square rooted to find the length of the hypotenuse. Consequently, the hypotenuse is the longest side in a right triangle.