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Need help with Triangle ABC

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by Baldini » Thu Mar 05, 2009 7:43 am
In triangle ABC (in attachment) what is the length of side BC?

(1) Line segment AD has length of 6
(2) x=36

OA is A

Could someone please explain to me how to get the answer?
Grazie mille
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Source: — Data Sufficiency |
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by DanaJ » Thu Mar 05, 2009 7:53 am
Let's start by analyzing the picture: notice that ADB = 180 - 2x, since the sum of angles ADB and BDC is 180. Now, similarly, ADB = 180 - BAD - DBA = 180 - x - DBA, since the sum of the angles of triangle ADB is 180. This means that:
180 - 2x = 180 - x - DBA, making DBA = x. Now, Since triangle ADB has two angles of the same length (DBA and BAD), it's an isosceles triangle, with AD = BD.
This is also true for triangle BCD: angles BDC and BCD are equal, meaning that triangle BCD is an isosceles triangle, with BD = BC.
Now use these two equalities to establish that AD = BD = BC or that AD = BC.

1 tells us that AD = 6, making BC = AD = 6 as well. So 1 is sufficient.
2 is insufficient because it does not provide any lengths.
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by Baldini » Thu Mar 05, 2009 8:45 am
thanks for that great explanation!
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