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

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

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

C) \[45{}^\circ \]

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

Correct Answer: C

Solution :

[c] According to the question, \[AB.BC=\frac{1}{2}.A{{C}^{2}}\] \[\Rightarrow \]   \[2\,AB.\,\,BC=A{{C}^{2}}\] \[\Rightarrow \]   \[2\,AB.\,\,\,BC=A{{B}^{2}}+B{{C}^{2}}\] \[\Rightarrow \]   \[A{{B}^{2}}+B{{C}^{2}}-2AB.\,\,BC=0\] \[\Rightarrow \]   \[{{(AB-BC)}^{2}}=0\]\[\Rightarrow \]\[AB=BC\] \[\therefore \]      \[\angle BAC=\angle BCA=45{}^\circ \]