Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)

Chapter 0: Tutorial; page 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.

Parts

1. Let's assume that one of the equal sides is the hypotenuse and the other is a:
   
   $$\begin{array}{l}c=s\\ a=s\end{array}\}$$


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.


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


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

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

Score

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