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

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


(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.


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Attachment:
2017-07-24_1034.png
2017-07-24_1034.png [ 9.32 KiB | Viewed 3233 times ]
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A
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broall
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ABCD is a quadrilateral, as shown in the diagram below. What is the [#permalink] 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?


(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.


Show 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.

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

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] 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 2706 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] 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?


(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.


Show 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.

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] New post   28 Feb 2019, 01:57
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.



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



Can you explain how you came up with the above part?
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the [#permalink] New post   02 Mar 2019, 08:59
ALso 16∗36=SAOD∗SBOC≤14(SAOD+SBOC)2⟹SAOD+SBOC≥48

Can you please explain this
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the [#permalink] New post   03 Mar 2019, 22:48
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.


Show 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.

The answer is A.



Although the answer is correct, but I feel there is one mistake in the formula used:

\((a+b)^2\)=\(a^2\)+\(b^2\)+\(2ab\)
=>\((a+b)^2\) > \(2ab\) {As squared term is always greater than 0}
=>\(ab\)<\(\frac{1}{2} (a+b)^2\)

So instead of \(\frac{1}{4}\), it should be \(\frac{1}{2}\) from which we get \((a+b)>24\sqrt{2}\).

Thus the minimum value as required is \(24\sqrt{2}\).
Please correct me if I am wrong. Thanks.
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the [#permalink] New post   30 Nov 2023, 11:16
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Re: ABCD is a quadrilateral, as shown in the diagram below. What is the [#permalink]
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