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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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23 Jul 2017, 22:35
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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?

(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.

[Reveal] Spoiler:
Attachment:

2017-07-24_1034.png [ 9.32 KiB | Viewed 657 times ]
[Reveal] Spoiler: OA

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

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24 Jul 2017, 01:35
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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?

(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.

[Reveal] Spoiler:
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.

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

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07 Nov 2017, 02:59
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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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07 Nov 2017, 04:24
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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

Posted from my mobile device

Posted from my mobile device
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the [#permalink]

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07 Nov 2017, 04:42
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?

(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.

[Reveal] Spoiler:
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.

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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07 Nov 2017, 06:29
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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 [ 9.59 KiB | Viewed 264 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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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the   [#permalink] 07 Nov 2017, 06:29
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