question_answer

AD is the median of a triangle ABC and 0 is the centroid such that AO = 10 cm. The length of OD in cm is

A)  4                    

B)  5

C)  6        

D)         8

Correct Answer: B

Solution :

        In \[\Delta \]ABC, \[\angle A=x,\angle B=y:\angle C=z\] In \[\Delta \] PBC \[\angle PBC+\angle PCB+\angle BPC=180{}^\circ \] \[\Rightarrow \]\[\frac{1}{2}\]\[\angle EBC+\angle FCB+2\angle BPC=360{}^\circ \] = \[180{}^\circ \] \[\Rightarrow \]\[\angle EBC+\angle FCB+2\angle BPC=360{}^\circ \] \[\Rightarrow \]\[\text{(180}{}^\circ -y)\text{ +}\left( 180{}^\circ -z \right)+2\text{ }\angle BPC\] =\[360{}^\circ \] \[\Rightarrow \]\[360{}^\circ -\left( y+z \right)+2\] \[\angle BPC=360{}^\circ \] \[\Rightarrow \]\[2\angle BPC=y+z\] \[\Rightarrow \]\[2\angle BPC=180{}^\circ -x\] \[=180{}^\circ -\angle BAC\] \[\therefore \]\[\angle BPC=90{}^\circ -\frac{1}{2}\angle BAC\] \[=90{}^\circ -50{}^\circ =40{}^\circ \]