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An isosceles trapezoid is inscribed in a semicircle as shown below, such that the three shaded regions are congruent. The radius of the semicircle is one meter. How many meters are in the perimeter of the trapezoid? Express your answer as a decimal to the nearest tenth.

 

 May 28, 2021
 #1
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If  the  shaded  regions  are equal.....then  the  chords  that comprise  the  top and  sides  of  the  trapezoid  must  be  equal

 

Call the  top two  vertices A and B   and  the center  0

 

Then . connecting  the top  two  vertices  to  the center   will  form  triangle  AOB  with  angle  AOB =  60°   and  OA, OB  =  1 m

 

Using  the  Law  of Cosines  we  can find   chord AB   as  :

 

AB^2   =  OA^2  + OB^2   -  2(OA * OB) cos (AOB)

 

AB^2  =  1^2  +  1^2   -  2 ( 1 * 1) (1/2)

 

AB^2   =  2 - 1

 

AB^2  =  1

 

AB =  1

 

The   perimeter of  the  trapezoid  =  3AB   +  diameter   =   3 (1)  +  2   =    5 m

 

 

cool cool cool

 May 29, 2021

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