question_answer

In a right angled triangle, the product of two sides is equal to half of the square of the third side, i.e. hypotenuse. One of the acute angles must be                                                       [SSC (Multitasking) 2014]

A) \[60{}^\circ \]                          

B) \[30{}^\circ \]

C) \[45{}^\circ \]  

D) \[15{}^\circ \]

Correct Answer: C

Solution :

According to the question,

\[AB\cdot BC=\frac{1}{2}\cdot A{{C}^{2}}\]

\[\Rightarrow \]   \[2AB\cdot BC=A{{C}^{2}}\]

\[\Rightarrow \]   \[2AB\cdot BC=A{{B}^{2}}+B{{C}^{2}}\]

\[\Rightarrow \]\[A{{B}^{2}}+B{{C}^{2}}-2AB\cdot BC=0\]

\[\Rightarrow \]   \[{{(AB-BC)}^{2}}=0\]

\[\Rightarrow \]               \[AB=BC\]

\[\therefore \]                  \[\angle BAC=\angle BCA=45{}^\circ \]