show that in a right angled triangles, the hypotenuse is the longest s...
Proof that the Hypotenuse is the Longest Side of a Right-Angled Triangle
A right-angled triangle is a triangle with one angle measuring 90 degrees. The longest side of a right-angled triangle is called the hypotenuse. To prove that the hypotenuse is the longest side, we need to use the Pythagorean Theorem.
Pythagorean Theorem
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that a, b, and c are the lengths of the sides of a right-angled triangle, where c is the hypotenuse.
According to the Pythagorean Theorem,
c² = a² + b²
Proof
To prove that the hypotenuse is the longest side in a right-angled triangle, we need to show that c is greater than both a and b.
Let's assume that a is the longest side. In this case, we have:
a² = b² + c²
Now, we can rearrange the equation to get:
c² = a² - b²
Since a is the longest side, a > b. Therefore, a² > b².
Hence, we have:
c² = a² - b² > 0
This means that c is greater than 0. Therefore, c > 0.
So, we have shown that c is greater than both a and b, which means that the hypotenuse is the longest side in a right-angled triangle.
Conclusion
From the above proof, we can conclude that in a right-angled triangle, the hypotenuse is always the longest side. This is because the square of the hypotenuse is equal to the sum of the squares of the other two sides, and as a result, the hypotenuse will always be greater than both of the other two sides.