 Hi and welcome to the session. Let us discuss the following question. Question says in figure 12.33 ABC is a quadrant of a circle of radius 14 centimeters and a semi-circle is drawn with BC as diameter. Find the area of the shaded region. This is the given figure 12.33. First of all let us understand that area of sector of a circle is equal to theta upon 360 multiplied by pi r square where r is the radius of the circle and theta is the angle of the sector. And area of triangle is equal to half into base into height. Also area of semi-circle is equal to pi r square upon 2 where r is the radius of the circle. Now we will use these formulas as our key idea to solve the question. Let us now start with the solution. Now we have to find area of the shaded region. We are given that ABC is a quadrant of a circle of radius 14 centimeters. So AC is equal to AB is equal to 14 centimeters. Clearly we can see AC and AB are radii of this circle. Now we will find area of quadrant ABC. We know area of sector is equal to theta upon 360 multiplied by pi r square where theta is the angle of the sector and r is the radius of the circle. Now clearly we can see quadrant is a sector where theta is equal to 90 degrees and radius is equal to 14 centimeters. So area of quadrant ABC is equal to 90 upon 360 multiplied by pi multiplied by square of 14 centimeters square. Now we will substitute 22 upon 7 for pi in this expression and we get area of quadrant ABC is equal to 90 upon 360 multiplied by 22 upon 7 multiplied by 14 centimeters square. Now simplifying we get area of quadrant ABC is equal to 154 centimeters square. Now to find the area of this shaded region we will subtract area of segment BC from area of this semicircle. Clearly we can see area of segment BC is equal to area of quadrant ABC minus area of triangle ABC. So we can write area of triangle ABC is equal to half multiplied by base multiplied by height. Now in triangle ABC right angle that A, AB is the height and AC is the base. So area of triangle ABC is equal to half multiplied by 14 multiplied by 14 centimeters square. Now multiplying these three terms we get area of triangle ABC is equal to 98 centimeters square. Now we know area of segment BC is equal to area of quadrant ABC minus area of triangle ABC. Now substituting corresponding values of area of quadrant ABC and area of triangle ABC in this expression we get area of segment BC is equal to 154 minus 98 centimeters square. Now subtracting these two terms we get 56 centimeters square. So we get area of segment BC is equal to 56 centimeters square. So clearly we can see area of this segment BC is equal to 56 centimeters square. Now we know that BC is the diameter of the given semicircle. Now first of all we will find out BC in right triangle ABC, BC square is equal to AB square plus AC square using Pythagoras theorem. We get hypotenuse square is equal to perpendicular square plus base square. Now substituting corresponding values of AB and AC in this expression we get square of BC is equal to 14 square plus 14 square. Now this implies BC square is equal to 196 plus 196. Now this implies square of BC is equal to 392. Taking square root on both the sides we get BC is equal to 14 multiplied by root 2 centimeters. We are given that BC is the diameter of the semicircle. So we get diameter of semicircle is equal to BC is equal to 14 root 2 centimeters. Now radius of the semicircle that is R is equal to half of diameter. So it is equal to 14 root 2 upon 2 centimeters which is further equal to 7 root 2 centimeters. Now from key idea we know area of semicircle is equal to Py R square upon 2. Now substituting corresponding value of R and Py in this expression we get area of given semicircle is equal to 22 upon 7 multiplied by 7 root 2 multiplied by 7 root 2 upon 2 centimeter square. Now simplifying this expression we get 11 multiplied by root 2 multiplied by 7 root 2 centimeter square. Multiplying these three terms we get 154 centimeter square. So we get area of given semicircle is equal to 154 centimeter square. Now we know area of shaded region is equal to area of given semicircle minus area of segment BC. So we can write area of shaded region is equal to area of given semicircle minus area of segment BC. Now substituting corresponding values of area of given semicircle and area of segment BC we get 154 minus 56 centimeter square. Now subtracting these two terms we get 98 centimeter square. So area of shaded region is equal to 98 centimeter square. So we can write area of shaded region is equal to 98 centimeter square. This is our required answer. This completes the session. Hope you understood the solution. Take care and keep smiling.