question_answer

P is the midpoint of\[\overline{\text{AB}}\]. M and N are midpoints of \[\overline{\text{AP}}\]respectively. What is the measure of\[\overline{\text{MN}}\]?

A) \[\frac{1}{3}\overline{\text{AB}}\]                         

B) \[\frac{1}{2}\overline{\text{AB}}\]  

C) \[\frac{1}{2}\overline{\text{AP}}\]   

D) \[\frac{3}{2}\overline{\text{AB}}\]

Correct Answer: B

Solution :

P is the midpoint of \[\overline{\text{AB}}\]\[\Rightarrow \text{AP=PB}\] M and N are midpoints of \[\overline{\text{AP}}\] and \[\overline{\text{PB}}\]\[\Rightarrow \]\[\overline{\text{AM}}\text{ = }\overline{\text{MP}}\] and \[\overline{\text{PN}}\text{ = }\overline{\text{NB}}\] \[\therefore \overline{\text{MN}}\text{=}\overline{\text{MP}}\text{+}\overline{\text{PN}}\]\[\text{=}\frac{\text{1}}{\text{2}}\overline{\text{AP}}\text{+}\frac{\text{1}}{\text{2}}\overline{\text{PB}}\text{=}\frac{\text{1}}{\text{2}}\left( \overline{\text{AP}}\text{+}\overline{\text{PB}} \right)\text{=}\frac{\text{1}}{\text{2}}\text{(}\overline{\text{AB}}\text{)}\]