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ABCD is a quadrilateral, as shown in the diagram below. What is the

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ABCD is a quadrilateral, as shown in the diagram below. What is the  [#permalink]

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New post 23 Jul 2017, 23:35
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ABCD is a quadrilateral, as shown in the diagram below. What is the minimum possible sum of areas of triangles AOD and BOC?
Image

(1) Area of triangles AOB and COD are 16 and 36, respectively.
(2) Area of triangle AOD is equal to the area of triangle BOC.


Attachment:
2017-07-24_1034.png
2017-07-24_1034.png [ 9.32 KiB | Viewed 931 times ]

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ABCD is a quadrilateral, as shown in the diagram below. What is the  [#permalink]

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New post 24 Jul 2017, 02:35
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Bunuel wrote:
ABCD is a quadrilateral, as shown in the diagram below. What is the minimum possible sum of areas of triangles AOD and BOC?
Image

(1) Area of triangles AOB and COD are 16 and 36, respectively.
(2) Area of triangle AOD is equal to the area of triangle BOC.


Attachment:
2017-07-24_1034.png


(1) We have \(S_{AOB} = 16\) and \(S_{COD} = 36\)

Also \(\frac{S_{AOB}}{S_{BOC}}=\frac{AO}{OC}\)
\(\frac{S_{AOD}}{S_{DOC}}=\frac{AO}{OC}\)

Hence \(\frac{S_{AOB}}{S_{BOC}} = \frac{S_{AOD}}{S_{DOC}} \\
\implies S_{AOD} * S_{BOC} = S_{AOB} *S_{DOC} = 16 * 36 \)

ALso \(16 * 36 = S_{AOD} * S_{BOC} \leq \frac{1}{4}( S_{AOD} + S_{BOC})^2 \implies S_{AOD} + S_{BOC} \geq 48\)

Sufficient.

(2) We have no more information about the area of AOD and BOC, hence we can't know the minimum possible sum of areas of triangles AOD and BOC. Insufficient.

The answer is A.
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the  [#permalink]

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New post 07 Nov 2017, 03:59
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Hi Broall

Thanks for your answer. Your mathematical notation was somehow not well interpreted by the system. Could you please edit if you a have a minute?

Many Thanks
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the  [#permalink]

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New post 07 Nov 2017, 05:24
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Octobre wrote:
Hi Broall

Thanks for your answer. Your mathematical notation was somehow not well interpreted by the system. Could you please edit if you a have a minute?

Many Thanks


Done. Thank you

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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the  [#permalink]

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New post 07 Nov 2017, 05:42
broall wrote:
Bunuel wrote:
ABCD is a quadrilateral, as shown in the diagram below. What is the minimum possible sum of areas of triangles AOD and BOC?
Image

(1) Area of triangles AOB and COD are 16 and 36, respectively.
(2) Area of triangle AOD is equal to the area of triangle BOC.


Attachment:
2017-07-24_1034.png


(1) We have \(S_{AOB} = 16\) and \(S_{COD} = 36\)

Also \(\frac{S_{AOB}}{S_{BOC}}=\frac{AO}{OC}\)
\(\frac{S_{AOD}}{S_{DOC}}=\frac{AO}{OC}\)

Hence \(\frac{S_{AOB}}{S_{BOC}} = \frac{S_{AOD}}{S_{DOC}} \\
\implies S_{AOD} * S_{BOC} = S_{AOB} *S_{DOC} = 16 * 36 \)

ALso \(16 * 36 = S_{AOD} * S_{BOC} \leq \frac{1}{4}( S_{AOD} + S_{BOC})^2 \implies S_{AOD} + S_{BOC} \geq 48\)

Sufficient.

(2) We have no more information about the area of AOD and BOC, hence we can't know the minimum possible sum of areas of triangles AOD and BOC. Insufficient.

The answer is A.



Hey,

What does S mean? (Area)
and how is
\(\frac{S_{AOB}}{S_{BOC}}=\frac{AO}{OC}\)
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the  [#permalink]

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New post 07 Nov 2017, 07:29
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Abhishiek wrote:
Hey,

What does S mean? (Area)
and how is
\(\frac{S_{AOB}}{S_{BOC}}=\frac{AO}{OC}\)


Yep, S mean the area.

Attachment:
2017-07-24_1034.png
2017-07-24_1034.png [ 9.59 KiB | Viewed 516 times ]


BH is perpendicular to AC.

We have
\(S_{AOB} = \frac{1}{2} * AO * BH\)
\(S_{BOC} = \frac{1}{2} * OC * BH\)

Hence
\(\frac{S_{AOB}}{S_{BOC}}=\frac{\frac{1}{2} * AO * BH}{\frac{1}{2} * OC * BH}=\frac{AO}{OC}\)

Hope this helps.
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Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the &nbs [#permalink] 07 Nov 2017, 07:29
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