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Given a semicircle with diameter 5, three squares are constructed as shown below.  Find the sum of the areas of the three squares.

 

 Dec 24, 2020
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Let  the eqaution of the  circle  be   x^2 + y^2   = 2.5^2   ⇒  x^2 + y^2   = 6.25

 

Call  the side of the  purple square, a

 

Then,  when y = a ,  we can find  x  as    sqrt ( 6.25  - a^2 )

 

And using symmetry,   the distance from  P  to this point  = 2 sqrt (6.25 - a^2)

 

Using the secant-tangent theorem, we  have  that

 

a^2  = [ (1/2)a ] [( (1/2)a  + 2sqrt ( 6.25 - a^2 ]

 

a^2=  (1/4)a^2  +  a sqrt (6.25 -a^2)

 

(3/4)a^2  =  a sqrt ( 6.25 -a^2)    divide out a

 

(3/4)a  = sqrt (6.25 - a^2)        square both sides

 

(9/16)a^2   = 6.25  - a^2

 

(9/16)a^2  =  25/4 - a^2

 

(25/16)a^2  =  100/16

 

25a^2   =100

 

a^2 = 100/25  =  4

 

a  = 2

 

 

So....the area of the pink square  = a^2   =  2^2   = 4

 

And the area of  the orange square is  [ (1/2) a]^2  = [(1/2)(2)]^2  =  1

 

And the area of  the   aqua suare is  ( 5 -2)^2   =3^2   = 9

 

So....the total area is    4 + 1 + 9  =   14

 

cool cool cool

 Dec 24, 2020

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