question_answer

\[\overrightarrow{\text{MN}}\] is the perpendicular bisector of \[\overleftrightarrow{\text{AB}}\]. Which of the given statements is correct?

(i) \[\angle \text{ANM+}\angle \text{MNB=90}{}^\circ \]

(ii) \[\overline{\text{AN}}\text{=}\overline{\text{NB}}\]

(iii) \[\overline{\text{AN}}\text{=2 }\overline{\text{NB}}\]

(iv) \[\angle \text{MNB=}\frac{1}{2}\angle \text{ANM}\]

A) (i) and (iii) only 

B) (ii) and (iv) only

C) (i) and (ii) only  

D) (ii) and (iii) only

Correct Answer: C

Solution :

\[\overrightarrow{\text{NM}}\bot \overleftrightarrow{\text{AB}}\] and \[\overrightarrow{\text{NM}}\] divides \[\overleftrightarrow{\text{AB}}\] into two congruent parts. Clearly \[\angle ANM\text{ }=\text{ }\angle MNB\text{ }=\text{ }90{}^\circ \] is true. \[\overline{\text{AN}}\text{=}\overline{\text{NB}}\] is true since \[\overrightarrow{\text{NM}}\bot \overleftrightarrow{\text{AB}}\] \[\overline{\text{AN}}\text{ = 2}\overline{\text{NB}}\] is false, and \[\angle \text{MNB=}\frac{1}{2}\angle \text{ANM}\] is false. Thus, only (i) and (ii) are correct.