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avatar+85 

Please solve these!
1.

2.

 Mar 8, 2024
 #1
avatar+128794 
+1

First one

 

Law of Cosines

 

(x + 4)^2 = x^2 + (x + 2)^2  - 2(x * (x + 2))cos 120°          (cos 120°    = -1/2)

 

x^2 + 8x + 16  =  x^2 + x^2 + 4x + 4 + x (x + 2)

 

x^2 + 8x + 16  =  2x^2 + 4x + 4 + x^2 + 2x

 

x^2 + 8x + 16 =  3x^2 + 6x + 4

 

2x^2 - 2x - 12 =   0

 

x^2 - x  - 6 =  0

 

(x - 3) ( x + 2)  = 0

 

x  = 3

 

cool cool cool          

 Mar 8, 2024
 #2
avatar+128794 
+1

Second one

 

Let N be the center of the circle   and its radius =  r

Let M be the tangent point of the circle to OA

OM  =  3

ON = 5 - r

MN =  r

 

We can  form  a right triangle  ONM such that

 

OM^2  + MN^2  =  ON^2

 

3^2 + r^2  =  (5-r)^2

 

3^2 + r^2  = r^2  - 10r + 25

 

10r  =  25 - 9

 

10r  = 16

 

r = 16 / 10  =  8/5 =   1.6

 

cool cool cool

 Mar 8, 2024

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