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Exercise 1
Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 20072020)
Chapter 0: Tutorial; page 1
Tutorial, Exercise 1
Question
A rightangle triangle has two identical sides of length s each. How long is the third side?
Reminder
... here is the Pythagorean theorem for a rightangle 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
 Let's assume that one of the equal sides is the hypotenuse and the other is a:
$$\begin{array}{l}c=s\\ a=s\end{array}\}$$
 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
$${c}^{2}={s}^{2}+{s}^{2}=2{s}^{2}$$
 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