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In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What

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In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post 09 Dec 2010, 06:58
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In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What is the length of DE?

(A) 6
(B) 7.5
(C) 8
(D) 9
(E) 10
[Reveal] Spoiler: OA

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Last edited by Bunuel on 09 Jul 2013, 09:50, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Geometry problem [#permalink] New post 09 Dec 2010, 08:53
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klevs1985 wrote:
Attachment:
The attachment prob2.pdf is no longer available


In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What is the length of DE?

(A) 6
(B) 7.5
(C) 8
(D) 9
(E) 10
Attachment:
untitled.PNG
untitled.PNG [ 7.97 KiB | Viewed 4040 times ]
As all inscribed angles that subtend the same arc are equal then <BCD=<BED (as these angles subtend the arc BD) and <CBE=<CDE (as these angles subtend the arc CE). Also <BAC=<DAE. So triangles ABC and ADE are similar: in similar triangles, corresponding sides are all in the same proportion.

So, DE/BC=AD/AB --> DE/6=6/4 --> DE=9.

Answer: D.
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Re: Geometry problem [#permalink] New post 09 Dec 2010, 20:09
Bunuel I looked at the difference between the two sides since they had to be in same proportion. So just took 6/4 AD/AB = 3/2 and since BC=6 then DE=3/2(6) = 9 is this correct way of looking at it? Also the triangles are mirror images of each other so therefor side BA=AD? how do we know these are the angles that are corresponding.
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Re: Geometry problem [#permalink] New post 10 Dec 2010, 00:14
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gettinit wrote:
Bunuel I looked at the difference between the two sides since they had to be in same proportion. So just took 6/4 AD/AB = 3/2 and since BC=6 then DE=3/2(6) = 9 is this correct way of looking at it? Also the triangles are mirror images of each other so therefor side BA=AD? how do we know these are the angles that are corresponding.


Because triangles ABC and ADE are similar their corresponding sides are all in the same proportion. Corresponding sides are the sides opposite the angles which are equal. For example as <BCA=<AED then the sides opposite them BA and AD are corresponding. So, in similar triangles the RATIO of corresponding sides are the same: BA/AD=BC/DE.

Next triangles are not congruent they are similar, so AB doesn't equal to AD (by the way stem gives AB = 4 and AD = 6).

For more check triangles and circles chapters of math book: math-triangles-87197.html and math-circles-87957.html

Hope it's clear.
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Re: Geometry problem [#permalink] New post 10 Dec 2010, 08:31
This makes sense now.
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Re: Geometry problem [#permalink] New post 05 May 2011, 06:45
DE/BC = AD/AB

=> DE = 6 * 6/4 = 9

Answer - D
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Re: Geometry problem [#permalink] New post 24 May 2014, 09:09
Bunuel wrote:
klevs1985 wrote:
Attachment:
prob2.pdf


In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What is the length of DE?

(A) 6
(B) 7.5
(C) 8
(D) 9
(E) 10
Attachment:
untitled.PNG
As all inscribed angles that subtend the same arc are equal then <BCD=<BED (as these angles subtend the arc BD) and <CBE=<CDE (as these angles subtend the arc CE). Also <BAC=<DAE. So triangles ABC and ADE are similar: in similar triangles, corresponding sides are all in the same proportion.

So, DE/BC=AD/AB --> DE/6=6/4 --> DE=9.

Answer: D.


Hi Bunuel,

I looked through the GMAT Club mathbook but i'm still having a hard time making the leap here:

What exactly does "subtend" mean? Why do you say that <BCD=<BED and not <BCD=<CDE (isn't it a transverse cutting a parallel line)?

Thanks!
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Re: Geometry problem [#permalink] New post 24 May 2014, 09:41
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Expert's post
russ9 wrote:
Bunuel wrote:
klevs1985 wrote:
Attachment:
The attachment prob2.pdf is no longer available


In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What is the length of DE?

(A) 6
(B) 7.5
(C) 8
(D) 9
(E) 10
Attachment:
The attachment untitled.PNG is no longer available
As all inscribed angles that subtend the same arc are equal then <BCD=<BED (as these angles subtend the arc BD) and <CBE=<CDE (as these angles subtend the arc CE). Also <BAC=<DAE. So triangles ABC and ADE are similar: in similar triangles, corresponding sides are all in the same proportion.

So, DE/BC=AD/AB --> DE/6=6/4 --> DE=9.

Answer: D.


Hi Bunuel,

I looked through the GMAT Club mathbook but i'm still having a hard time making the leap here:

What exactly does "subtend" mean? Why do you say that <BCD=<BED and not <BCD=<CDE (isn't it a transverse cutting a parallel line)?

Thanks!


Angles BCD and BED are based on minor arc BD, thus they are equal:
Attachment:
a.png
a.png [ 7.37 KiB | Viewed 448 times ]


Check for more here: math-circles-87957.html
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Geometry problem   [#permalink] 24 May 2014, 09:41
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