Krishna
0

Step 1: Set up a condition to get the area of the shaded region

           Area of shaded shape = area of quadrilateral AMOP - area of sector MOP 


Step 2:  Find the area of the quadrilateral

          NOTE:  Quadrilateral AMOP is made up of two congruent right triangles (AC perpendicular to PO and AB perpendicular to MO) since P and M are points of tangency. (Note: AP = a/2) 

                 Area of the quadrilateral = 2* area of the right triangle  

               Skill 1:  Find the area of the right triangle

                             \frac{1}{2} (b * h)

               Skill 2: Find the base and height of the right triangle

                          Base b = \frac{side}{2}   = \frac{10}{2}

                          Height h = radius of the inscribed circle = side *\frac{\sqrt{3}}{6}

                         h = 10 * \frac{\sqrt{3}}{6}

                Skill 3: Plugging the base and height values in the formula

                             \frac{1}{2} *\frac{10}{2}*10 * \frac{\sqrt{3}}{6}

                             25 \frac{\sqrt{3}}{6}

                So, area of the quadrilateral = 2* 25 \frac{\sqrt{3}}{6}

          

Step 3:Calculate the area of the sector

            NOTE: Find the angle of the sector = \frac{360}{3} = 120 \degree

            

          [Step 1: Recall the area of the sector formula  

            NOTE: \frac{\theta}{360} * \pi r^2


            Step 2: Substitute all the values in the formula.

          EXAMPLE:   \frac{\theta}{360} * \pi r^2

                        \frac{120}{360}\ *\ 3.14\ *\ \left(10\ \frac{\sqrt{3}}{6}\right)^2


            Step 3: Simplify the equation

            EXAMPLE:  \frac{\pi}{3}\cdot\ \left(10\frac{\sqrt{3}}{6}\right)^2

                                = \frac{50\ \pi}{18}

            

Step 4:  Calculate the area of the shaded region

             NOTE: Area of the shaded region = area of quadrilateral AMOP - area of sector MOP 

                   Area of the shaded region = 2* 25 \frac{\sqrt{3}}{6} - \frac{50 \pi}{18}

                                                             Simplify for the answer