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# help with geo!

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Chords AB and CD of a circle are parallel, and angle BDC = angle ACD = 45 degrees.  If CD = 10 and the radius of the circle is 6, then find AB.

Dec 11, 2020

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AB = √44

Dec 11, 2020
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Let M  be  the  midpoint of  the  chord of 10

Let the center of the  circle   =  (0,0)  = O

The equation of the circle is x^2 + y^2   = 36   (1)

Connect  OC, OM

OC   =6, MC  =5

We can find OM thusly

sqrt  ( OC^2  -MC^2)  =  sqrt  (6^2  -5^2)  = sqrt (11)

Let  the coordinates of  C = (5, -sqrt (11) )

The  slope of the line containing segment AC  = -1

The equation of this line  is    y = -(x -5)  -sqrt (11)

y+ x  = 5  -sqrt (11)      square both sides

x^2 + 2xy  + y^2 =  36 - 10sqrt (11)   sub (1)  into this

36 + 2xy  = 36 -10 sqrt (11)

2xy = -10sqrt (11)

xy = -5sqrt (11)

y = -5sqrt (11)/x

Sub  this into (1)  to find the  x cooordinate  of  A

x^2  +  ( -5sqrt (11)/x)^2  =36

x^2  + 275/x^2  = 36       multiply through  by  x^2

x^4  + 275  = 36x^2

x^4  - 36x^2  +  275  =  0

Factor  as

(x^2   - 25)(x^2 -11)   = 0

The solutions are   x  = -5, 5, -sqrt (11), sqrt (11)

The x coordinate of A  cannot be -5 ...so  the x coordinate of  A  =   -sqrt (11)

And  due to symmetry,  the  x  coordinate of  B  =sqrt (11)

So

AB = sqrt (11)  -  -sqrt (11)  =   2sqrt (11)

Dec 13, 2020