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, |
\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\] |
\[={{(a+c)}^{2}}-2ac\] |
\[={{(56-b)}^{2}}-2\times 168\] |
\[\Rightarrow \] \[{{b}^{2}}=3136+{{b}^{2}}-112b-336\] |
\[\Rightarrow \] \[112b=2800\] |
\[\Rightarrow \] \[b=\frac{2800}{112}=25\] |
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