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805+ (Hard)|   Geometry|      
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Bunuel
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ABCD is a quadrilateral, as shown in the diagram below. What is the 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.


Show Spoiler
Attachment:
2017-07-24_1034.png
2017-07-24_1034.png [ 9.32 KiB | Viewed 3920 times ]
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A
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ABCD is a quadrilateral, as shown in the diagram below. What is the 24 Jul 2017, 02:35
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Bunuel
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
Show more

(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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ABCD is a quadrilateral, as shown in the diagram below. What is the 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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ABCD is a quadrilateral, as shown in the diagram below. What is the 07 Nov 2017, 05:24
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Octobre
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
Show more

Done. Thank you

Posted from my mobile device

Posted from my mobile device
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ABCD is a quadrilateral, as shown in the diagram below. What is the 07 Nov 2017, 05:42
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Bunuel
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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Hey,

What does S mean? (Area)
and how is
\(\frac{S_{AOB}}{S_{BOC}}=\frac{AO}{OC}\)
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ABCD is a quadrilateral, as shown in the diagram below. What is the 07 Nov 2017, 07:29
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Abhishiek

Hey,

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

Yep, S mean the area.

Attachment:
2017-07-24_1034.png
2017-07-24_1034.png [ 9.59 KiB | Viewed 3300 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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ABCD is a quadrilateral, as shown in the diagram below. What is the 28 Feb 2019, 01:57
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Bunuel
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 \(16 * 36 = S_{AOD} * S_{BOC} \leq \frac{1}{4}( S_{AOD} + S_{BOC})^2 \implies S_{AOD} + S_{BOC} \geq 48\)

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

Can you please explain this
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ABCD is a quadrilateral, as shown in the diagram below. What is the 03 Mar 2019, 22:48
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broall
Bunuel
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.
Show more


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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ABCD is a quadrilateral, as shown in the diagram below. What is the 30 Nov 2023, 11:16
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