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In right triangle ABC, the length of side \overline{AC} is 8, the length of side \overline{BC} is 6, and \angle C = 90^\circ. The circumcircle of triangle ABC is drawn. The angle bisector of \angle ACB meets the circumcircle at point M. Find the length CM.

 Mar 22, 2020
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See the image below :

 

 

Let A  =(8,0)

Let B = (0,6)

Let C = (0,0)

 

The  center  of the circumcenter  lies at the  midpoint of the hypotenuse = (4,3)

And the radius  of the ctrcumcircle =   5

 

So.....the equation of the circumcircle = 

 

(x - 4)^2  + (y-3)^2  = 25  (1)

 

And since  angle ACB  = 90°  then  the angle bisector will be  a   45°  line through the origin

 

And  the  equation  of  this line  is  y  = x

 

Substiting  this into (1)  for x gives the  x coordinate of  M

 

So

 

(x - 4)^2  + ( x -3)^2  = 25

 

x^2 -8x + 16   + x^2  -6x + 9   = 25    simplify

 

2x^2  - 14x =  0

 

2x ( x - 7)  =  0

 

Setting each factor to 0 and solving for x  produces  x  =  0     or  x   = 7

 

We  want the second result

 

So   the y coordinate of M   =  7

 

So....the length of  CM =   sqrt  (7^2 + 7^2)  =  7sqrt(2) units  

 

 

cool cool cool

 Mar 22, 2020

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