 Hello and welcome to the session. In this session we discuss the following question which says, A, B, C, D is the diameter of a circle of radius 4.5 centimeters. The lengths A, B, C and C, D are equal. Semi-circles are drawn with A, B and B, D as diameters. If A, D is equal to 9 centimeters, find the perimeter and area of the shaded region. Take pi equal to 3.14. So this is the figure given to us. Here this A, B, C, D is the diameter of a circle which has radius 4.5 centimeters. And taking A, B and B, D as the diameters, we have drawn two semi-circles. And we have to find the perimeter and area of this shaded region. Let us recall the formula. For the perimeter of a circle, this is equal to 2 pi r where this r is the radius of the circle and the area of a circle is equal to pi r square where again r is the radius of the circle. So this is the key idea that we use in this question. Now let us move on to the solution. We are given that A, D is equal to 9 centimeters and we are also given that A, B is equal to B, C is equal to C, D. So we have A, D is equal to, from the figure you can see A, B plus B, C plus C, D. And you know that these three parts are equal. So this would mean that A, B is equal to, B, C is equal to, C, D is equal to 9 upon 3 centimeters. That is equal to 3 centimeters. From the figure we have that B, D is equal to A, D minus A, B. Now A, D is equal to 9 centimeters as given to us and A, B as we have found out is equal to 3 centimeters. So 9 centimeters minus 3 centimeters is equal to 6 centimeters. That is we have B, D is equal to 6 centimeters. Now let us find out the perimeter of the shaded region and this would be equal to arc A, E, B plus arc B, F, D plus A, G, D. Now since we know that the perimeter of a circle is 2 pi r, so the perimeter of a semicircle would be 2 pi r upon 2, that is pi r. So now perimeter of the semicircle A, E, B, that is arc A, E, B is given by pi into its radius. Now since A, B is equal to 3 centimeters that is we have the diameter A, B is 3 centimeters so its radius would be 3 upon 2 plus arc B, F, D that is perimeter of arc B, F, D which is given by pi into the radius of the semicircle drawn with B, D as the diameter. Now as B, D is equal to 6 centimeters so its radius would be 6 upon 2 plus the arc A, G, D that is the semicircle drawn with A, D as the diameter as A, D is equal to 9 centimeters so the perimeter of this A, G, D semicircle is given by pi into the radius which is 9 upon 2. So this centimeter is the perimeter of the shaded region and this is equal to 3 pi plus 6 pi plus 9 pi this whole upon 2 centimeters which is further equal to 18 pi upon 2 centimeters and this is equal to 9 pi centimeters is the perimeter of this shaded region. Now taking value of pi as 3.14 we get this is equal to 28.26 centimeters is the perimeter of the shaded region. Now let's find out the area of the shaded region and this is equal to the area of semicircle A, E, B plus area of the semicircle A, G, D that is the semicircle minus the area of the semicircle B, F, D. As you know area of a circle is given by pi R square so area of a semicircle would be given by pi R square upon 2. Now first let's see what would be the area of the semicircle A, E, B. This is the semicircle A, E, B. Now A, B is equal to 3 centimeters as you know so its radius would be 3 upon 2 centimeters so 1 upon 2 into pi R square that is 3 upon 2 whole square plus the area of the semicircle A, G, D. This is the semicircle A, G, D. Now this A, D is given as 9 centimeters and we are given that the radius of the circle is 4.5 centimeters that is 9 upon 2 so area of the semicircle A, G, D is given as half into pi into 9 upon 2 whole square minus the area of the semicircle B, F, D that is the semicircle. We know that B, D is equal to 6 centimeters that is the diameter for the semicircle B, F, D is 6 centimeters and so its radius is 6 upon 2 centimeters and hence area of the semicircle B, F, D is 1 upon 2 into pi into 6 upon 2 whole square and so this is equal to pi upon 2 common into 3 upon 2 whole square plus 9 upon 2 whole square minus 6 upon 2 whole square centimeter square is the area of the shaded region this is equal to pi upon 2 into 9 upon 4 plus 81 upon 4 minus 6 upon 4 centimeter square this is equal to pi upon 2 into 9 plus 81 minus 36 upon 4 centimeter square further we get pi upon 2 into 54 upon 4 centimeter square now 2, 27 times is 54 so this is equal to 27 upon 4 pi centimeter square putting the value for pi we get 27 upon 4 into 3.14 centimeter square is the area of the shaded region and this is equal to 21.195 centimeter square that is the area of the shaded region is equal to 21.195 centimeter square and we also got the perimeter of the shaded region equal to 28.26 centimeters so this is our final answer this completes the session hope you have understood the solution of this question