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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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This makes sense now.
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DE/BC = AD/AB

=> DE = 6 * 6/4 = 9

Answer - D
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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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Bunuel
klevs1985
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 24830 times ]

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