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In a parallelogram ABCD, P is the midpoint of AD. Also, BP and AC intersect at Q. Then AQ : QC =
1. 1 : 3
2. 3 : 1
3. 2 : 1
4. 1 : 2

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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

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