+4609
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
\(a+b+c\)?
+128474 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
