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# geometry help!!!!!1

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The triangle shown is an equilateral triangle with side length 12 cm. A side of the triangle is the diameter of the circle. If the sum of the areas of the two small shaded regions in square centimeters in simplest radical form is $$a\pi - b\sqrt{c}$$, what is

$$a+b+c$$? Apr 1, 2018

#1
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Note...tertre....if we find the center of the circle and draw a radius  parallel to the bottom base of the large equilateral triangle, we will create another smaller equilateral triangle with a side of 6 cm

So.....the area of the sector formed by the two radial sides of this smaller equilateral triangle  will equal 1/6 of the circle whose radius  = 6  =

(1/6)pi (6^2)  =  6pi  cm^2        (1)

And the area of this smaller  equilateral triangle =

(1/2)(6^2)*sqrt (3) / 2  =

36sqrt (3) /4  =

9sqrt (3)  cm ^2       (2)

So....the   area of one of the shaded regions  is   (1)  - (2)  =

{ 6pi - 9sqrt (3)]  cm^2

So...by symmetry.....both shaded regions have an area of

2 [ 6pi - 9sqrt (3)]  cm^2  =

[ 12pi  - 18sqrt (3) ]    cm^2

So

a + b + c  =

12 + 18 + 3

33   Apr 1, 2018
edited by CPhill  Apr 1, 2018
#2
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Wow, your amazing CPhill! Thank you!

tertre  Apr 1, 2018