# In a right triangle, is the side opposite the right angle the shortest side?

Dec 28, 2015

No. It is the longest side. It can be shown by using the Sines Theorem.

#### Explanation:

We will show that the side opposite to the right angle is the longest side in a right triangle.

According to the Sine Theorem for every triangle we have:

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

Let's assume, that angle $C$ is right. So the last fraction is equal to $c$ (the denominator becomes $1$), so we can write that:

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

So we can calculate $a$ and $b$ in terms of side $c$

$a = c \cdot \sin A$ and $b = c \cdot \sin B$

$A$ and $B$ are accute angles, so we can write, that sinA in (0;1) and sinB in (0;1), so if $\sin A < 1$ then $a = c \cdot \sin A < c$

In the same way we can show that $b = c \cdot \sin B < c$, so finally both $a$ and $b$ are smaller than $c$ which concludes the proof.