]> Exercise 1

# Tutorial, Exercise 1

## Question

A right-angle triangle has two identical sides of length s each. How long is the third side?

## Reminder

... here is the Pythagorean theorem for a right-angle triangle, where c is the length of the hypotenuse, and the symbols a and b - of the other sides.

 ${c}^{2}={a}^{2}+{b}^{2}$ (0.0.1.1)

## Parts

1. Let's assume that one of the equal sides is the hypotenuse and the other is a:

$\begin{array}{l}c=s\\ a=s\end{array}\right\}$

2. In this case, from (0.0.1.1) we obtain that

$b=0$

or in other words this is not a triangle, in contradiction to the requirement.

3. The assumption $\begin{array}{l}c=s\\ b=s\end{array}\right\}$ yields a similar result.

4. The only solution is therefore: $\begin{array}{l}a=s\\ b=s\end{array}\right\}$
5. The substitution in (0.0.1.1) yields

${c}^{2}={s}^{2}+{s}^{2}=2{s}^{2}$
6. and finally $c=\sqrt{2}s$

## Score

Parts 1,2,4,5 are worth 2 points each.

Parts 3,6 are worth 1 point each