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Let A and B be points on a circle of radius 2 centered at O, such that angle AOB = 90.  C is the midpoint of OB, and AC extended meets the circle at D.  Find the length CD.

 Jul 2, 2020
 #1
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See the following  image  :

 

 

 

 

 

Let O = (0,0)

Let A = (2,0)

Let B = (0,2)

Let E = (-2,0)

 

C =(0,1)

 

Note that BE  is the diameter  = 4

And  CE  = BE - BC  = 4 - 1  =   3

 

Since AOB is right  with OA =  2   and  OC =  1

Then, by the Pythagorean Theorem  AC  =sqrt  ( OA^2  + OC^2)  = sqrt  ( 2^2 + 1^2)  =  sqrt (5)

 

By the intersecting chord theorem  we  have that

 

BC * CE  =  AC * CD

 

1 * 3  =  sqrt (5) * CD

 

3 = sqrt (5) * CD

 

CD =  3/sqrt (5)  ≈  1.34

 

 

cool cool cool

 Jul 2, 2020

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