question_answer

If the two angle on the base of a triangle are \[{{\left( 22\frac{1}{2} \right)}^{o}}\] and \[{{\left( 112\frac{1}{2} \right)}^{o}}\], then the ratio of the height of the triangle to the length of the base is               [MP PET 1993; Pb CET 2002]

A) 1 : 2

B) 2 : 1

C) 2 : 3

D) 1: 1

Correct Answer: A

Solution :

In \[\Delta ACD,\,\frac{h}{\sin {{67.5}^{o}}}=\frac{AC}{\sin {{90}^{o}}}\] \[\Rightarrow \frac{h}{AC}=\sin {{67.5}^{o}}\]....(i) In\[\Delta ABC,\,\frac{AC}{\sin {{22.5}^{o}}}=\frac{x}{\sin {{45}^{o}}}\Rightarrow \frac{AC}{x}=\sqrt{2}\sin {{22.5}^{o}}\] .....(ii) From (i) and (ii), \[\frac{h}{x}=\frac{1}{2}\].