Question

In △ABC, CM is the median to AB and side BC is 12 cm long. There is a point P∈ CM and a line AP intersecting BC at point Q. Find the lengths of segments CQ and BQ , if P is the midpoint of CM

Answer 1

4 cm

Explanation:

Draw the line MD that is parallel to AQ. Consider triangles BMD and BAQ. These triangles are similar, because they have three pairs of congruent angles (angles BMD and BAQ are corresponding, angles BDM and BQA are corresponding and angle B is common). The coefficient of similarity is
`(1)/(2),` because point M is midpoint of AB. Then

`(BD)/(BQ)=(1)/(2)\Rightarrow BQ=2BD.`

Since
`BQ=BD+DQ,` then
`DQ=2BD-BD=BD.`

Consider triangles CPQ and CMD. These triangles are similar, because they have three pairs of congruent angles (angles CPQ and CMD are corresponding, angles CQP and CDM are corresponding and angle C is common). The coefficient of similarity is
`(1)/(2),` because point P is midpoint of CM. Then

`(CQ)/(CD)=(1)/(2)\Rightarrow CD=2CQ.`

Since
`CD=CQ+DQ,` then
`DQ=2CQ-CQ=CQ.`

Hence you get
`CQ=DQ=BD=(BC)/(3)=(12)/(3)=4\ cm.`