In the shape shown below, A, B and C are the vertices of an equilateral triangle, side length xcm. The arc AC forms a semicircle. Which of the following alternatives is a correct expression for the area, in cm^2, of the shape?

Answer:

21) A

The third side of the equilateral triangle is the diameter of the semi-circle.

Therefore, diameter of semi-circle = ‘x’ cm

Radius = ½x = (ˣ⁄₂)

We first need to find the area of the equilateral triangle. In order to do this, we need to find its height using Pythagoras theorem.

Divide the equilateral triangle into two right-angled triangles.
Hypotenuse = ‘x’
1st side = ½x
2nd side (height) = A

x² = (½x)² + A²
x² – ¼x² = A²
¾x² = A²
A = √(¾x²) = x√3
                       2

Area of equilateral triangle = 2 x area of right-angled triangle

Area of 1 right-angled triangle =  1  * (x√3) *  x   =  x²√3
                                                     2       2        2         8

Area of equilateral triangle = 2 * x²√3  =  x²√3 
                                                       8            4

Area of semi-circle = ½πr²

  1   * π *   x²   =  πx²
  2              2²         8

Total area = area of triangle + area of semi-circle

x²√3  +   πx²
  4             8

2x²√3  +   πx² 
          8

2x²√3  +   πx² 
         8

x² (2√3  +   π) 
           8          

So the answer is A