question_answer

The perimeter of a certain isosceles right triangle is \[10+10\sqrt{2}\,cm.\] What is the length of the hypotenuse of the triangle?

A)  5 cm              

B)  10 cm

C)  \[5\sqrt{2}\,\,cm\]      

D)       \[10\sqrt{2}\,\,cm\]

Correct Answer: B

Solution :

  \[AB=BC=x\text{ }cm.\] \[\therefore \]\[AC=\sqrt{A{{B}^{2}}+B{{C}^{2}}}=\sqrt{{{x}^{2}}+{{x}^{2}}}=\sqrt{2{{x}^{2}}}\] \[=\sqrt{2}x\,cm.\] \[\therefore \]   \[AB+BC+CA=10+10\sqrt{2}\] \[\Rightarrow \]            \[2x+\sqrt{2}x=10+10\sqrt{2}\] \[\Rightarrow \]            \[\sqrt{2}x\,\,\left( \sqrt{2}+1 \right)=10\left( 1+\sqrt{2} \right)\] \[\Rightarrow \]            \[\sqrt{2}x=10\,cm.=AC\]