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Analytic geo, perhaps?

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This problem is very intresting, as I first got the correct answer using analytic geo, but wondered if there was another, easier way to do this.

 

  If ΔBCD is isosceles and ㄥADC = 105° and BC = 5√6, find AC in the figure below:

 

https://ibb.co/1GfxMNx

 Dec 18, 2020

5+0 Answers

 #1
avatar+1641 
-3

If ΔBCD is isosceles and ㄥADC = 105° and BC = 5√6, find AC.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

∠ADB = 1/2 (105º)       ∠ABD = 90º - ∠ADB           AB = BC

 

AC = 2 (sin∠ABD * 5√6)

 

 Dec 18, 2020
 #2
avatar+421 
+1

Please be careful when you are answering, ABD is not half of 105 degrees and AB≠AC.       

 Dec 18, 2020
edited by Pangolin14  Dec 18, 2020
edited by Pangolin14  Dec 18, 2020
edited by Pangolin14  Dec 18, 2020
 #4
avatar+1641 
+4

Sorry about that!  For some reason, I read ΔBCD as ABCD. cheeky

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

BC = CD = 5√6          ∠CBD = ∠BDC = 45º          ∠ADB = 105 - 45 = 60º

 

BD = 10√3            AD = cos(60º) * 10√3 = 5√3

 

AC = 16.73  smiley

 

jugoslav  Dec 18, 2020
edited by jugoslav  Dec 18, 2020
edited by jugoslav  Dec 18, 2020
 #5
avatar+421 
0

Thanks! Exactly what I got!

Pangolin14  Dec 20, 2020
 #3
avatar
+3

Angle CBA =75 degrees
Angle CBD =45 degrees
Angle DBA =30 degrees

 

By Pythagoras
Side AD =5sqrt(3)
Side AB =15
Side BD =10sqrt(3)
Side BC =5sqrt(6)

 

Using Law of Cosines:
Side AC =16.73

 Dec 18, 2020

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