Prove that in a right angled triangle, hypotenuse is the longest side.

To prove that in a right-angled triangle, the hypotenuse is the longest side, we can use the Pythagorean theorem and logical reasoning.

Step 1: Understand the triangle
Consider a right-angled triangle ABC, where ∠C = 90°, and the sides are as follows:

AB = hypotenuse (the side opposite the right angle),
BC = one leg,
AC = the other leg.
Step 2: Apply the Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle:

AB² = AC² + BC²

Here, AB is the hypotenuse, and AC and BC are the legs of the triangle.

Step 3: Compare the sides
To prove that the hypotenuse is the longest side, we need to show that AB > AC and AB > BC.

From the Pythagorean theorem, AB² = AC² + BC².
Both AC² and BC² are positive numbers, so AC² + BC² is greater than either AC² or BC² alone. Therefore, AB² > AC² and AB² > BC².
Since AB² is greater than both AC² and BC², it follows that AB > AC and AB > BC (because the square root function preserves the inequality for positive numbers).
Step 4: Conclusion
Thus, we have shown that the hypotenuse (AB) is greater than both of the legs (AC and BC), proving that in a right-angled triangle, the hypotenuse is the longest side.