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In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...

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In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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New post Updated on: 30 Jan 2019, 08:14
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In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC intersects PD at Q. What proportion of AC is AQ?
(A) 1/2
(B) 1/3
(C) 1/4
(D) 1/5
(E) 1/6

Attachment:
Screen Shot 2019-01-25 at 3.50.29 PM.png
Screen Shot 2019-01-25 at 3.50.29 PM.png [ 78.65 KiB | Viewed 726 times ]


Thank you in advance!

Source: Manhattan Review

Originally posted by jpfg259 on 25 Jan 2019, 07:54.
Last edited by jpfg259 on 30 Jan 2019, 08:14, edited 1 time in total.
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Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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New post 25 Jan 2019, 11:06
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jpfg259 wrote:
In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC intersects PD at Q. What proportion of AC is AQ?
(A) 1/2
(B) 1/3
(C) 1/4
(D) 1/5
(E) 1/6

Attachment:
The attachment Screen Shot 2019-01-25 at 3.50.29 PM.png is no longer available


Thank you in advance!



Triangles APQ and CDQ are similar triangles as AP is parallel to CD...
The corresponding sides are in same ratio...
so \(\frac{AP}{CD}=\frac{PQ}{DQ}=\frac{AQ}{CQ}\) means \(\frac{AP}{CD}=\frac{AQ}{CQ}... =>\frac{x}{2x}=\frac{AQ}{CQ}....=>CQ=2AQ\)..
We are looking for \(\frac{AQ}{AC}=\frac{AQ}{AQ+QC}=\frac{AQ}{AQ+2AQ}=\frac{AQ}{3AQ}=\frac{1}{3}\)
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Screen Shot 2019-01-25 at 3.50.29 PM.png
Screen Shot 2019-01-25 at 3.50.29 PM.png [ 61.58 KiB | Viewed 674 times ]


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Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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New post 29 Jan 2019, 08:05
chetan2u wrote:
jpfg259 wrote:
In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC intersects PD at Q. What proportion of AC is AQ?
(A) 1/2
(B) 1/3
(C) 1/4
(D) 1/5
(E) 1/6

Attachment:
Screen Shot 2019-01-25 at 3.50.29 PM.png


Thank you in advance!



Triangles APQ and CDQ are similar triangles as AP is parallel to CD...
The corresponding sides are in same ratio...
so \(\frac{AP}{CD}=\frac{PQ}{DQ}=\frac{AQ}{CQ}\) means \(\frac{AP}{CD}=\frac{AQ}{CQ}... =>\frac{x}{2x}=\frac{AQ}{CQ}....=>CQ=2AQ\)..
We are looking for \(\frac{AQ}{AC}=\frac{AQ}{AQ+QC}=\frac{AQ}{AQ+2AQ}=\frac{AQ}{3AQ}=\frac{1}{3}\)


That was a great explanation, thank you so much!
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Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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New post 05 Feb 2019, 09:48
jpfg259 wrote:
In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC intersects PD at Q. What proportion of AC is AQ?
(A) 1/2
(B) 1/3
(C) 1/4
(D) 1/5
(E) 1/6

Attachment:
Screen Shot 2019-01-25 at 3.50.29 PM.png


Thank you in advance!

Source: Manhattan Review


Important thing to note in this question is that triangles QDC & QPA are similar. Since P is the mid point; AP:DC is 1:2; Hence AQ:QC is 1:2 or AQ:AC is 1:3

Corect ans is Option B
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Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...   [#permalink] 05 Feb 2019, 09:48
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