 Hi and welcome to the session. Let us discuss the following question. Question says, the area of an equilateral triangle ABC is 17320.5 cm2. With vertex of triangle as center, a circle is drawn with radius equal to the half the length of the sites of triangle. Find the area of the shaded region. This is the given figure 12.28. First of all, let us understand that area of equilateral triangle is equal to root 3 upon 4 multiplied by site square and area of sector is equal to theta upon 360 multiplied by pi r square where theta is the angle of sector in degrees and r is the radius of the circle. Now we will use these formulas as our key idea to solve the given question. Let us now start with the solution. We are given that triangle ABC is equilateral triangle and these three are sectors. Now we have to find area of this shaded region. So if we subtract area of these three sectors from area of triangle ABC, then we get area of shaded region. Now we are given that area of triangle ABC is equal to 17320.5 cm2. Now we know radius of each circle is equal to half the length of the site of the triangle. Now we will find out site of the triangle first. From key idea we know area of equilateral triangle is equal to root 3 upon 4 multiplied by site square. Now let us assume that each site of equilateral triangle ABC is equal to x cm. Now we can substitute x for site in this formula and we get root 3 upon 4 x square is equal to 17320.5 cm2. Now multiplying both sides by 4 upon root 3 we get x square is equal to 17320.5 multiplied by 4 upon root 3. Now substituting value of root 3 given in the question in this expression we get x square is equal to 17320.5 multiplied by 4 upon 1.73205. We are given root 3 is equal to 1.73205. So simplifying further we get 173205 upon 10 multiplied by 4 0 0 0 0 upon 173205. This term will get cancelled by this term and 0 will cancel 0 and we get x square is equal to 40000. Now taking square root on both the sides we get x is equal to 20000. So we get x is equal to 200 cm or we can say each side of equilateral triangle ABC is equal to 200 cm. Now we know radius of each circle is equal to half the length of the side of the equilateral triangle ABC. So we get radius of circle that is r is equal to 200 upon 2 cm which is further equal to 100 cm. Now we have to find area of these 3 sectors. We know radius of the circle is equal to 100 cm. Now we will find out angle of the sectors. Now we know this is the equilateral triangle and in equilateral triangle all angles are equal. So by angles and property of triangle we get each of the angle of this triangle is equal to 60 degrees. So we can write angle A is equal to angle B is equal to angle C is equal to 60 degrees. Now we get angle of each sector is equal to 60 degrees. Also we know area of a sector is equal to theta upon 360 multiplied by pi r square where theta is the angle of the sector in degrees and r is the radius of the circle. We will find area of a given sector. We know theta is equal to 60 degrees and r is equal to 100 cm. So we get area of a given sector is equal to 60 upon 360 multiplied by pi multiplied by square of 100 cm square. Now we will substitute value of pi given in the question that is 3.14. So we get area of a given sector is equal to 60 upon 360 multiplied by 3.14 multiplied by 100 multiplied by 100 cm square. Now we can find area of 3 similar sectors by multiplying this expression by 3. So we can write area of 3 given sectors is equal to 3 multiplied by 60 upon 360 multiplied by 3.14 multiplied by 100 multiplied by 100 cm square. Now simplifying this expression we get 15700 cm square is equal to area of 3 given sectors. Now we can find area of the shaded region by subtracting area of these 3 sectors from area of equilateral triangle ABC. So we get area of shaded region is equal to area of equilateral triangle ABC minus area of 3 sectors. Now substituting corresponding value of area of equilateral triangle ABC and area of 3 given sectors, we get 17320.5 minus 15700 cm square is equal to area of shaded region. Now this is further equal to 1620.5 cm square. So we get area of the shaded region is equal to 1620.5 cm square. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.