# How is the hypotenuse the longest side of any right triangle?

I see that the hypotenuse of a right triangle is opposite the right angle, but how is it always the longest side? I also know that it connects to endpoints of other sides. Please help me out with this! I'm really wanting to know this surprising thing. Here's an example of a right triangle:

This is an isosceles right triangle because sides a and b (the height and the base) are the same lengths with two of the angles being 45 degrees adding up to a total with the right angle of 180 degrees (all triangles have angles that add up to 180 degrees). I just want to know from this triangle or any other right triangles why the hypotenuse is the longest side. You'll really be helping me out.

• Pythagorean Theorem. – Edward Jiang Oct 26 '14 at 2:18
• Wow. I know that $$a^2+b^2=c^2$$ and I'm hoping what you said is good. – Mathster Oct 26 '14 at 2:20
• It's the result of a basic theorem in Euclidean Geometry: if in the triangle $\;\Delta ABC\;$ we have that $\;AB>AC\;$ , then $\;\angle C>\angle B\;$ , and in words: given two sides of a triangle and their opposite angles, the biggest angle is opposite to the biggest side... and the other way around . The claim now follows from the easy fact that in euclidean geometry a triangle can have at most one straight angle, which automatically is then the biggest one. – Timbuc Oct 26 '14 at 4:15

Let $a$ be the hypotenuse and $b,c$ the others sides, then by Pythagorean Theorem $$a^2=b^2+c^2.$$ Then $$a^2>b^2,\quad a^2>c^2.$$ Therefore $$a>b\quad a>c.$$

• This is fundamental, but it does not seem like an intuitive thought – Eduardo S. Jun 25 '18 at 1:38

Alternatively, you have the law of sines: For any triangle with sides, $A,B,C$ and corresponding angles $a,b,c$ with angle $a$ opposite side $A$ et cetera, you have the following:

$$\frac{\sin a}{A} = \frac{\sin b}{B} = \frac{\sin c}{C}$$

Let $a$ be $90$ degrees, making $A$ our hypotenuse. Since $a+b+c = 180$ and we don't want to consider negative angles or angles equal to zero in a triangle, we have that $b<90$ and $c<90$.

$\sin a = \sin (90^\circ) = 1$

$0 < \sin b < 1$ for $0^\circ<b<90^\circ$

Using these pieces of information, you have that $B\cdot\sin a = B = A\cdot \sin b < A$

Showing that $B < A$.

Similar proof shows that $C < A$.

All of this together shows that the side opposite the 90 degree angle is the longest side in the triangle.

In a similar fashion, you can show that even for triangles which are not right triangles, the side opposite the biggest angle will be the biggest side.

If the angles are $\angle A \le \angle B \lt \angle C$,
then the sides are $a \le b \lt c.$
On a right triangle, two angles are acute and the third is a right angle.
Hence the side opposite the right angle is the longest side.