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

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klevs1985
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In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   Updated on: 09 Jul 2013, 10:50
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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
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D

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prob2.pdf [9.83 KiB]
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Originally posted by klevs1985 on 09 Dec 2010, 07:58.
Last edited by Bunuel on 09 Jul 2013, 10:50, edited 1 time in total.
Renamed the topic and edited the question.
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Bunuel
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Re: Geometry problem [#permalink] New post   09 Dec 2010, 09:53
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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 31895 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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gettinit
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Re: Geometry problem [#permalink] New post   09 Dec 2010, 21:09
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, 01:14
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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, 09:31
This makes sense now.
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Re: Geometry problem [#permalink] New post   05 May 2011, 07:45
DE/BC = AD/AB

=> DE = 6 * 6/4 = 9

Answer - D
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Re: Geometry problem [#permalink] New post   24 May 2014, 10:09
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, 10:41
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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 22975 times ]


Check for more here: math-circles-87957.html
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Re: In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   20 Feb 2016, 11:23
angle A is at the intersection of two lines..thus, the angle A of the both triangles must be the same. somewhere I read that the angles of the cord when represented on the circle are the same...so triangles must be similar.
AD is similar to AB. the multiplier factor thus must be 6/4 or 3/2. now, we know that BC is 6. ED will be 6*3/2 or 9.
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Re: In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   06 Jan 2017, 23:49
Equal triangles will have same proportions of sides of the triangles.

DE/BC = AD/AB

DE = 6 * 6/4 = 9. My answer is D.
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Re: In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   16 Jul 2020, 02:09
Bunuel yashikaaggarwal why would it be wrong to solve this by chord-chord theorem?
6*5=4*DE
30/4=DE
7.5=DE

could you please assist?
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Re: In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   16 Jul 2020, 02:18
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Kritisood wrote:
Bunuel yashikaaggarwal why would it be wrong to solve this by chord-chord theorem?
6*5=4*DE
30/4=DE
7.5=DE

could you please assist?


It should be:
BA*AE = DA*AC
4*AE = 6*5
AE = 15/2
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Re: In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   16 Jul 2020, 03:49
Kritisood wrote:
Bunuel yashikaaggarwal why would it be wrong to solve this by chord-chord theorem?
6*5=4*DE
30/4=DE
7.5=DE

could you please assist?

Alternatively you can check by similar triangle theorem.
Both triangles are similar,
ABC and ADE
CASE 1: Let say AC is similar to AD
So AB is similar to AE
BC is Similar to DE
X = DE
So 5/6 = 6/x
5x = 36
X = 7.2 (WE DON'T HAVE 7.2 IN OPTION)

CASE 2: Let say AC is similar to AE
So AB is similar to AD
BC is Similar to DE
X = DE
So 4/6 = 6/x
4x = 36
X = 9
Which is one of the options we have.

Hence we can say it's our answer.

Posted from my mobile device
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Re: In the above circle AB = 4, BC = 6, AC = 5 and AD = 6. What [#permalink] New post   16 Aug 2023, 16:55
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