question_answer

For an isosceles right - angled triangle having each of equal sides 'x', We have

I. Area\[=\frac{1}{2}{{x}^{2}}\]

II. Perimeter\[=\left( 2+\sqrt{2} \right)x\]

III. Hypotenuse\[=2x\]

Which of the following is true?

A)  I only                          

B)  II only

C)  I and II                       

D)  I and III

Correct Answer: C

Solution :

Consider \[\Delta \text{ }ABC\] in which \[\angle B=90{}^\circ \]and AB = BC. Then, we must have     

I.   Area\[=\frac{1}{2}\times BC\times AB=\frac{1}{2}{{x}^{2}}.\]

II. \[AC=\sqrt{{{x}^{2}}+{{x}^{2}}}=\sqrt{2{{x}^{2}}}=\sqrt{2x}\]

\[\therefore \]Perimeter

\[=x+x+\sqrt{2x}=2x+\sqrt{2}x=x\left( 2+\sqrt{2} \right)\]

III. \[AC=\sqrt{2x}\ne 2x\]

\[\therefore \]I and II are true