question_answer

If the perimeter of a right angled triangle is 56 cm and area of the triangle is \[84\,\,c{{m}^{2}},\]then the length of hypotenuse is (in cm)                                                                                                                                         [SSC (10+2) 2012]

A) 25        

B) 50    

C) 7                                 

D) 24

Correct Answer: A

Solution :

Let a, b and c be the sides of the triangle and 6 is the length of the hypotenuse.

Now,     perimeter = 56

\[a+b+c=56\]

Area = 84

\[\Rightarrow \]   \[\frac{1}{2}\times ac=84\]

\[\Rightarrow \]   \[ac=168\]

By Pythagorus theorem,

\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\]

\[={{(a+c)}^{2}}-2ac\]

\[={{(56-b)}^{2}}-2\times 168\]

\[\Rightarrow \]   \[{{b}^{2}}=3136+{{b}^{2}}-112b-336\]

\[\Rightarrow \]   \[112b=2800\]

\[\Rightarrow \]   \[b=\frac{2800}{112}=25\]