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

Question

https://gmatclub.com/forum/download/file.php?id=13556 
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

Bunuel · Accepted Answer

untitled.PNG As all inscribed angles that subtend the same arc are equal then DE/6=6/4 --> DE=9. Answer: D.

gettinit · Answer

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.

Bunuel · Answer

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 math-triangles-87197.html and math-circles-87957.html Hope it's clear.

tfincham86 · Answer

This makes sense now.

subhashghosh · Answer

DE/BC = AD/AB

=> DE = 6 * 6/4 = 9

Answer - D

russ9 · Answer

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!

Bunuel · Answer

Angles BCD and BED are based on minor arc BD, thus they are equal: a.png Check for more here: math-circles-87957.html

mvictor · Answer

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.

prashantrchawla · Answer

Equal triangles will have same proportions of sides of the triangles.

DE/BC = AD/AB

DE = 6 * 6/4 = 9. My answer is D.

Kritisood · Answer

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?

Bunuel · Answer

It should be:
BA*AE = DA*AC
4*AE = 6*5
AE = 15/2

yashikaaggarwal · Answer

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

bumpbot · Answer

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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

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