Question

Find the obtuse angle between the medians of an isosceles right angled triangle drawn from the vertices of the acute angles.

• A. $\cos \theta =-4/5$
• B. $\sin \theta =-4/5$
• C. $\cos \theta =4/5$
• D. $\sin \theta =4/5$

Answer 1

The correct option is A $\cos \theta =-4/5$

With $O$ as origin and $OA=OB=\lambda$

then $A$ is $\lambda i$, $B$ is $\lambda j$ and $G$ is $\frac{\lambda }{3}(i+j).$

We have to find angle between medians $GA$ and $GB$ drawn from acute angles $A$ and $B.$

Median $\overrightarrow{GA}$ is $\lambda i-\frac{\lambda }{3}(i+j)=\frac{\lambda }{3}(2i-j)\Rightarrow \left|\overrightarrow{GA}\right|=GA=\frac{\lambda }{3}\sqrt{5}$

Median $\overrightarrow{GB}=\frac{\lambda }{3}(2j-i)\Rightarrow \left|\overrightarrow{GB}\right|=GB=\frac{\lambda }{3}\sqrt{5}$

$\overrightarrow{GA}.\overrightarrow{GB}=GA.GB.\cos \theta$

$\frac{{\lambda }^2}{9}(2i-j).(2j-i)=\frac{\lambda }{3}\sqrt{5}.\frac{\lambda }{3}\sqrt{5}\cos \theta$

$\Rightarrow -2-2=5\cos \theta \Rightarrow \cos \theta =-\frac{4}{5}$ etc.