Understanding the Hypotenuse in Right-Angled Triangles: A Comprehensive Guide

In a right-angled triangle, the hypotenuse is always the longest side. This fundamental property is a direct consequence of the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In a right-angled triangle, this theorem combined with the properties of a right angle leads to the hypotenuse being the longest side.

Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. For a right-angled triangle with sides a, b, and c, where c is the hypotenuse, the theorem implies:

Given that c is opposite the right angle (90 degrees), it must be the longest side because the other two angles are acute (less than 90 degrees).

Pythagoras' Theorem and the Hypotenuse

Pythagoras' Theorem provides a more direct method to demonstrate why the hypotenuse is the greatest side in a right-angled triangle. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

Mathematical Proof Using Pythagoras' Theorem

Consider a right-angled triangle with sides a, b, and c, where c is the hypotenuse:

$c^2  a^2   b^2$

Since (a^2), (b^2), and (c^2) are all positive numbers, we can deduce that:

$c^2 > a^2 quad text{and} quad c^2 > b^2$

Therefore, we conclude that:

$c > a quad text{and} quad c > b$

This mathematical proof clearly shows that the hypotenuse is always greater than either leg of the right-angled triangle.

Additional Insights on Right-Angled Triangles

In any triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle. In a right-angled triangle, the angle opposite the hypotenuse is the right angle (90 degrees), and the other two angles are acute, meaning they are less than 90 degrees. Since the sum of angles in a triangle is always 180 degrees, the angles in a right-angled triangle can vary but will always be less than or equal to 90 degrees.

Diameter of a Semi-Circle and the Hypotenuse

Interestingly, the hypotenuse of a right-angled triangle is also the diameter of a semi-circle. If you draw a semi-circle and select any point on its circumference, the line segment connecting that point to the two endpoints of the diameter forms two right-angled triangles. The diameter of the semi-circle is the hypotenuse of these triangles and is always the longest side. This property is a consequence of the Inscribed Angle Theorem, which states that an angle inscribed in a semicircle is a right angle.

Thus, if you draw any right-angled triangle with its hypotenuse as the diameter of a semi-circle, the angle subtended at the circumference by the diameter is always a right angle, and the hypotenuse is the largest side among all possible right-angled triangles with that diameter.