question_answer

Right angle triangle whose hypotenuse is of length p cm, one side of length q cm and p - q= 1 then the length of third side of the triangle is

A) \[\sqrt{1+2q}\,cm\]

B) \[\sqrt{p}+\sqrt{q}\,cm\]

C) \[\sqrt{p-q}\,cm\]

D) \[\sqrt{p}-\sqrt{q}\,cm\]

Correct Answer: A

Solution :

Let the third side be x cm.

\[{{P}^{2}}={{q}^{2}}+{{x}^{2}}\] [by Pythagoras theorem]

\[\Rightarrow \,\,\,{{x}^{2}}={{p}^{2}}-{{q}^{2}}\]

\[\Rightarrow \,\,\,{{x}^{2}}=\left( p-q \right)\left( p+q \right)\]

\[\Rightarrow \,\,\,\,x=\sqrt{\left( p+q \right)}\]       \[\left[ \because \,\,\,p=q=1 \right]\]

\[\Rightarrow \,\,x=\sqrt{\left( 1+q \right)+q}\]     \[\left[ p=1+q \right]\]

\[\Rightarrow \,\,\,\,x=\sqrt{2q+1\,}\,cm\]