Please help me just wrote an exam and wanted to know whether it was correct or not. I said $(m+n)(m-n)=(m-n)^2$ so I think it is not a right angled triangle. What do you guys say?

Hint

If it is a right triangle, you would have $a^2+b^2=c^2$. Here you are given $a=m^2-n^2$,$b=2mn$,$c=m^2+n^2$. Replace and see if it holds.

If it is a right-angled triangle then it must satisfy the Pythagorean theorem, so all what you have to do is to check whether: $$(2mn)^2+(m^2-n^2)^2\overset{\displaystyle ?}=(m^2+n^2)^2.$$ (By expanding the expressions in both sides then checking after simplifying if they are equal)

• I tried it and it doesn't work so that means that it isn't right angle – Jayshal rama Jun 17 '14 at 11:23
• @Jayshalrama Please recheck your work, it is actually equal! – Hakim Jun 17 '14 at 11:31

@claude is correct. It works out this way:

Out of $m^2+n^2,m^2-n^2,2mn,$ the number $m^2+n^2$ is the greatest. If it is a right-angled triangle, then $(m^2+n^2)^2 = (m^2-n^2) + (2mn)^2 m^4+n^4+2m^2\cdot n^2 = m^4+n^4-2m^2\cdot n^2 + 4m^2\cdot n^2 = m^4+n^4+2m^2\cdot n^2$ LHS = RHS hence triangle is right-angled by converse of pythagoras thm.

these 3 numbers also form a pythagorean triplet