Correct Answer - Option 4 : 1 : 2
Concept:
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Calculations:
Given, In a parallelogram ABCD, P is the midpoint of AD. Also, BP and AC intersect at Q.
Join AC and BP.
⇒ \(\angle\)AQP= CQB and \(\angle\)APQ= \(\angle\)CBQ
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent
⇒ \(\rm\triangle APQ ∼ \triangle CBQ\)
⇒ \(\rm \dfrac {AP}{BC}= \rm \dfrac{AQ}{QC}\)
⇒ AD = BC
⇒ \(\rm \dfrac {AP}{BC}= \rm \dfrac{AP}{AD} = \dfrac 1 2\)
⇒ \(\rm\rm \dfrac{AQ}{QC} = \dfrac 12\)
Hence, In a parallelogram ABCD, P is the midpoint of AD. Also, BP and AC intersect at Q. Then AQ : QC = 1 : 2