+0  
 
+1
126
2
avatar+2766 

 

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 

\(a+b+c\)?

 

tertre  Apr 1, 2018
 #1
avatar+87629 
+3

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

 

 

cool cool cool

CPhill  Apr 1, 2018
edited by CPhill  Apr 1, 2018
 #2
avatar+2766 
+2

Wow, your amazing CPhill! Thank you!

tertre  Apr 1, 2018

13 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.