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

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Joined: 22 Feb 2018
Posts: 10
In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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Updated on: 30 Jan 2019, 07:14
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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 [ 78.65 KiB | Viewed 441 times ]

Thank you in advance!

Source: Manhattan Review

Originally posted by jpfg259 on 25 Jan 2019, 06:54.
Last edited by jpfg259 on 30 Jan 2019, 07:14, edited 1 time in total.
Math Expert
Joined: 02 Aug 2009
Posts: 7334
Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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25 Jan 2019, 10:06
2
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 $$\frac{AQ}{AC}=\frac{AQ}{AQ+QC}=\frac{AQ}{AQ+2AQ}=\frac{AQ}{3AQ}=\frac{1}{3}$$
Attachments

Screen Shot 2019-01-25 at 3.50.29 PM.png [ 61.58 KiB | Viewed 394 times ]

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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

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Joined: 22 Feb 2018
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Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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29 Jan 2019, 07:05
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!
Director
Joined: 11 Feb 2015
Posts: 719
Re: In a parallelogram ABCD, P is the midpoint of AB. Diagonal AC inter...  [#permalink]

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05 Feb 2019, 08:48
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, 08:48
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