 Hello and welcome to the session. In this session, we discussed the following question that says, the radii of two concentric circles are 9 centimeters and 3 centimeters. AB is a diameter of bigger circle, BD is a tangent to the smaller circle, touching it at D, find the length of AB. Before moving on to the solution, let's recall one result which says that the tangent at any point of a circle and the radius through this point are perpendicular to each other. This is the key idea that we use for this question. Let's proceed with the solution now. We are given the two concentric circles with center O. We are given in this question that AB is the diameter of the bigger circle and this BD is the tangent to the smaller circle and we are supposed to find the length of AD. First of all, we join AE and OD. We have joined AE and OD. Now in the smaller circle, BD is the tangent of the circle and OD is the radius of the circle through the point D. Therefore, OD would be perpendicular to BE. Since we know that the tangent at any point of a circle and the radius through this point are perpendicular to each other. Now this BE is the chord of a bigger circle and this OD is the perpendicular from the center of the circle to this chord BE. Therefore, BD is equal to DE. Since we know that perpendicular from the center of the circle to a chord bisects the chord. Now since we get that BD is equal to DE, therefore this means that D is the midpoint of BE. Also we have that O is the midpoint of AB since AB is the diameter of the bigger circle. Now consider the triangle ABE in this O is the midpoint of side AB, D is the midpoint of side BE. Now since we know that the segment joining the midpoints of any two sides of a triangle is half the third side, therefore OD would be equal to half of AE. Since OD is the segment joining the midpoints of the two sides AB and BE of this triangle AEB, so this would be equal to half of the third side which is AE. We are given that the radius of two concentric circles are 9 cm and 3 cm, so this means that the radius of the bigger circle that is OB or OA would be equal to 9 cm and radius of the smaller circle would be equal to OD and that is 3 cm. Now that we have OD equal to half of AE, this means that AE would be 2 times of OD or you can say that AE is equal to 2 into 3 cm which would be equal to 6 cm thus we get AE equal to 6 cm. Now let us consider the right triangle ODB in this OB square is equal to OD square plus BD square that is BD square is equal to OB square minus OD square. This gives us BD equal to square root of OB square minus OD square. Now we have OB equal to 9 cm and OD equal to 3 cm, so we put the values here, so we get this equal to square root of 9 square minus 3 square that is equal to the square root of 81 minus 9 or you can say this is equal to square root of 72 which is equal to 6 root 2 thus we get BD is equal to 6 root 2 cm. Now since BD is equal to DE therefore we say that BD equal to DE is equal to 6 root 2 cm also AB is the diameter of the bigger circle therefore this angle AEB would be equal to 90 degrees since we know that angle in a semicircle is of measure 90 degrees and this angle AEB is angle in a semicircle thus this is also 90 degrees angle. Let us now consider the right triangle AED in this AD square is equal to AE square plus ED square we know that ED or say DE is 6 root 2 cm and AE is equal to 6 cm so this means that AD square is equal to 6 square plus 6 root 2 whole square or AD square is equal to 36 plus 72 which is equal to 108 this gives us AD is equal to square root of 108 and this is equal to 6 root 3 therefore we get AD is equal to 6 root 3 cm or you can say that AD is equal to 10.392 approximately cm thus we have obtained the length of AD this is our final answer this completes the session hope you have understood the solution of this question