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# In the figure above, BD = DC. If AD = 6, what is BD?

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In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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17 Jan 2019, 23:13
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65% (hard)

Question Stats:

59% (01:44) correct 41% (01:40) wrong based on 56 sessions

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In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Attachment:

2019-01-18_1111.png [ 6.46 KiB | Viewed 894 times ]

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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 12:06
could anybody explain it pls?

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In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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Updated on: 19 Jan 2019, 12:32
1
Easy one, <BAC = <2BDC because they subtend from the same Arc BC so <a is 10 now <ADB=160
Again <ADB+ <DAB + <DBA=180 <DBA= 180-160-10=10 ,we can now see that triangle BDA is an isosceles we two equal sides 10 .Therefore length must be equal AD=DB=6
Or better still look at my alternate solution down below

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Originally posted by Staphyk on 18 Jan 2019, 13:07.
Last edited by Staphyk on 19 Jan 2019, 12:32, edited 2 times in total.
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 14:24
Excuse me but how <a can be 20 if <ADB=180-20=160?? then <ABD is 0???
<BAC = <BDC only if BA||BD

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Posts: 243
Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 18:04
This should be changed from sub-600. I have no idea how to get the answer and don't understand Staphyk 's explanation
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 19:04
1
Bunuel wrote:

In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Will just place an idea here, on how i got B( A long explanation, Nevertheless an easy question)

If BD=DC, this implies triangle BDC is an isosceles triangle with < DBC = < BCD = 80

If one can prove triangle BDC ~ triangle ABC, we can use the ratio of sides to calculate the side BD
I am able to find, < C which will be common for both triangles, side BC which will be common again
Now do note i am missing out at one more factor to make it similar.

Nevertheless, now when i say those two triangles are similar, there is a property of similar triangles that ratio of corresponding sides are equall

I agree on the part that < ABD cant be zero, it will be 10( because < BDA = 160 and if you drop a perpendicular at AB < DAE = < DBE = 10.

Now if you understood what i wrote here, so i will be taking the corresponding sides to the equal angles
triangle BDC ~ triangle ABC

Do look into the properties of similar triangles

Since DC = BD

We can write BD = 6
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In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 19:42
My mind has just exploded)))..
Perpendicular droped on AB is not necessarily devides <ADB/2, therefore we can not assume that <DAE=<DBE=10...
Anyway sum of this two angles is 20 and answer should be close 6))

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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 19:53
1
Maikosh wrote:
My mind has just exploded)))..
Perpendicular droped on AB is not necessarily devides <ADB/2, therefore we can not assume that <DAE=<DBE=10...
Anyway sum of this two angles is 20 and answer should be close 6))

Posted from my mobile device

Good to know, that it somehow helped you in thinking how to solve the question.

I agreeon that part that if i drop a perpendicular bisector, only the sides will be equal.
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If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 22:59
Sorry you couldn’t get it Let’s me see wether I can still help
Now we given that line DC=BD if that is the case then <DCB=<DBC = (say x) now let’s solve x+x+20=180 x=80 therefore <DCB=80 and <DBC=80 also <ADB=160 since it’s supplementary to <BDC =20
Now rotate the figure clockwise such that Line BA sits flat we get <CBD to be complementary to <DBA=10 now triangle DBA adds up to 180 and we know <ADB=160 and <DBA=10 then <DAB must be 10 showing that line DA=DB =6 Hope it helps
Maikosh wrote:
Excuse me but how <a can be 20 if <ADB=180-20=160?? then <ABD is 0???
<BAC = <BDC only if BA||BD

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In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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18 Jan 2019, 23:05
1
How do you know that ∠ABC is 90º for ∠DBC and ∠DBA to be complimentary?
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In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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Updated on: 19 Jan 2019, 04:09
1
Bunuel wrote:

In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Attachment:
The attachment 2019-01-18_1111.png is no longer available

Imo b

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Attachments

IMG_20190119_173610277.jpg [ 1.03 MiB | Viewed 486 times ]

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Originally posted by Archit3110 on 19 Jan 2019, 02:35.
Last edited by Archit3110 on 19 Jan 2019, 04:09, edited 2 times in total.
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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19 Jan 2019, 03:59
Archit3110 wrote:
Bunuel wrote:

In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Attachment:
2019-01-18_1111.png

Imo b

Posted from my mobile device

In your solution, triangle ABD has a sum of 200, which cannot the case.
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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19 Jan 2019, 04:04
Archit3110 wrote:
Bunuel wrote:

In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Attachment:
2019-01-18_1111.png

Imo b

Posted from my mobile device

In your solution, triangle ABD has a sum of 200, which cannot the case.

it would be 10 ... not 20 ...
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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19 Jan 2019, 17:34
1
Bunuel wrote:

In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Attachment:
2019-01-18_1111.png

I feel this question is missing something. From the given info we cannot determine BD.
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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20 Jan 2019, 03:16
Staphyk Just want to clarify ,are we assuming a circumcircle .Then AC would become the diameter and the D would become the center and BD would also be the radius (also as isosceles triangle the angles would be 10 )
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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20 Jan 2019, 06:52
1
Bunuel wrote:

In the figure above, BD = DC. If AD = 6, what is BD?

A 4
B 6
C 8
D 10
E 12

Attachment:
The attachment 2019-01-18_1111.png is no longer available
Attachment:

2019-01-18_1111.png [ 7.7 KiB | Viewed 242 times ]

After finding the angles of the triangle you will get 2 Triangles ADB and BCD in which both the Triangles are Isosceles Triangles with a common vertex BD.

Thus in this figure AD = BD = CD = 6 , Answer will be (B) 6
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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20 Jan 2019, 15:47
Abhishek009 wrote:
Bunuel wrote:

After finding the angles of the triangle you will get 2 Triangles ADB and BCD in which both the Triangles are Isosceles Triangles with a common vertex BD.
Thus in this figure AD = BD = CD = 6 , Answer will be (B) 6

How do you know that ∠ABC is 90º for ∠DBC and ∠DBA to be complimentary?
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In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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20 Jan 2019, 22:24
Please explain how to find the other two angles to prove Triangle BAD is isosceles.
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Re: In the figure above, BD = DC. If AD = 6, what is BD?  [#permalink]

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20 Jan 2019, 22:51
Something is definitely missing in this question. Bunuel sir can you please check/ provide OE? Thanks.

BDC = 20 only gives us the info that the sum of the other two small angles is 20. ( Remote interior angles of triangle are same as exterior angle)

Also, just one of the two triangles is isoceles. So we cannot determine whether the split is 10-10 ( in which case the answer will be 6) or something else.

This is my take on it.

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Re: In the figure above, BD = DC. If AD = 6, what is BD?   [#permalink] 20 Jan 2019, 22:51
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