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Let AB be a diameter of a circle, and let C be a point on the circle such that AC = 8 and BC = 6. The angle bisector of < ACB intersects the circle at point M. Find CM.

 

This has been asked a couple times, specifiically here: http://web2.0calc.com/questions/help-please-due-tommorow-incredibly-hard

Although the answer may be correct, they use trig, which was not intended for this problem. 

Hints: AB = 10, Use Angle Bisector theorem, and power of a point(aka instersecting chords theorem, as CPhill calls it). 

 

Any help is appreciated! smiley

 Feb 4, 2021
 #1
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I used coordinates, and found C = (0,0) and M = (8,8), so CM = 8*sqrt(2).

 Feb 4, 2021
 #4
avatar+1209 
+3

Try these coordinates:   C (0, 0)        M (7, 7)                                                                                                                 

 

 CM = √98 = 7√2  smiley

jugoslav  Feb 4, 2021
edited by jugoslav  Feb 8, 2021
 #2
avatar+1209 
+4

https://web2.0calc.com/questions/question-help_7   (Very clever!!!) Bravo, MaxWong!!!

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

BC => a = 6         AC => b = 8           AB => c = 10

 

∠A = tan-1(6 / 8)             

 

∠ANC = 180 - (45 + ∠A)

 

By using the Law of Sines we can find side AN of triangle ACN.

 

AN ≈ 5.714           BN = 10 - AN ≈ 4.286

 

CN can be calculated by using this formula: CN = sqrt [ab(a + b + c)(a + b - c)] / a + b

 

CN ≈ 4.849 

 

By using the Intersecting chords theorem we can calculate the length of CM.

 

MN * CN = AN * BN             MN ≈ 5.05

 

CM = CN + MN ≈ 9.899  smiley

  

 Feb 4, 2021
edited by jugoslav  Feb 4, 2021
 #5
avatar+118505 
0

Max does get the correct answer....and BMA is a right triangle.....but I don't see how he arrives at the  conclusion that this triangle is isosceles with BA   =  MA

 

The only way that would be  possible is that if CM  were a diameter  (which it isn't)

 

Am I misssing something,  jugoslav  ????

 

cool cool cool

CPhill  Feb 4, 2021
edited by CPhill  Feb 4, 2021
 #6
avatar+1209 
+4

Hope that my diagram will answer your question.

 

Scale 1 : 1  There are a lot of different methods to calculate CM.

 

jugoslav  Feb 4, 2021
edited by jugoslav  Feb 8, 2021
 #3
avatar+118505 
+1

Per jugoslav's diagram.....Let N be the intersection of the  chords BA and MC

 

Since  BCA is  bisected, we  have the following relationship

 

BC /CA  = BN /AN

 

6/8 =  BN /AN  = 3/4

 

So   BN  =  (3/7) 10  =  30/7

And  AN  = (4/7)10  = 40/7

And  AN * BN =  1200 / 49

 

Without re-inventing the  wheel  the length of the  angle  bisector is  given by

 

BC * CA sqrt (2)  /  ( BC + CA)  =   *8*6 sqrt (2)  / ( 8 + 6) =  48sqrt (2) / 14  =  24sqrt (2)  /  7  = CN

 

So

 

AN * BN  =   CN  *   NM

 

1200/49  = 24sqrt (2) / 7  * NM

 

1200/ 49  *   7  / (24 sqrt (2))  =  NM =   25sqrt (2)  / 7   

 

Therefore    CN +  MN  =    CM  =   (25 + 24) sqrt (2)  / 7  =    49sqrt (2)  /7  =   7sqrt (2) 

 

 

 

 

Here's the proof of the length of the  angle bisector  :

 

 

 

 

cool cool cool

 Feb 4, 2021

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