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Right triangle $ABC$ has $AB = AC = 6$ cm. Circular arcs are drawn with centers at $A, B$ and $C,$ so that the arc centered at $A$ is tangent to side $BC$ and so that the arcs centered at $B$ and $C$ are tangent to the arc centered at $A,$ as shown. What is the perimeter of the shaded region? Express your answer as a decimal to the nearest hundredth.

 

 Aug 30, 2023
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The triangle is  isosceles so angles B and C  = 45°

 

The large  quarter circle  will be tangent  to BC at its midpoint =  (3,3)

 

The radius of this  circle =  sqrt (3^2 + 3^2)  = sqrt (18)  =  3sqrt (2)

 

The arc length of this  circle is   (1/4)  *2 * pi * ( 3sqrt (2))  =  (3/2)sqrt (2)  pi            (1)

 

The radius of each of the smaller arcs at B and C  =  6 - (3)sqrt (2)  

 

And the combined length of the arcs at B and C =  (1/4) * 2pi * ( 6 - 3sqrt 2)  =  

3pi  - (3/2)sqrt (2)pi      (2)

 

And the length of the shaded area bordering the hypotenuse  =  sqrt ( 6^2 + 6^2)  - 2 (6 - 3sqrt 2)  = 

6sqrt2 - 12 + 6sqrt 12  =   12 sqrt 2  - 12        (3)

 

The total perimeter of the  shaded area = (1) + (2) + (3)  = 

 

(3/2)sqrt( 2) pi   + 3pi - (3/2)sqrt  (2) pi  + 12sqrt (2) - 12  =

 

3pi + 12(sqrt 2 - 1)    = 14.395 cm ≈  14.4 cm

 

cool cool cool

 Aug 30, 2023

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