 Hello and welcome to the session. In this session, we discussed the following question which says, A, B and C are three points on a circle. The tangent at C meets BA produced at T. Given that angle ATC is equal to 36 degrees and angle ACT is equal to 48 degrees, calculate the angle subtended by AB at the center of the circle. Before moving on to the solution, let's discuss one result which says, 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 that AB and C are three points on a circle and the tangent at C meets BA produced at T. So this is a circle with center O and ABC are three points on the circle. The tangent at the point C meets BA produced at this point T. It's also given to us that angle ATC is of measure 36 degrees and angle ACT is of measure 48 degrees. Let's see what all we have in this question. We have that O is the center of the circle. Then CT is the tangent at point C. Then we have angle ACT is equal to 48 degrees and angle ATC is equal to 36 degrees. Now we are supposed to calculate the angle subtended by AB at the center of the circle. This is AB and we have to find out the angle subtended by this AB at the center of the circle. So for this, first of all, we would join OA. So this means we are supposed to find out this angle that is angle AOB. The CT is the tangent at the point C of the circle and OC is the radius through the point C. Therefore this angle between this tangent and the radius that is this angle, which is angle OCT would be of measure 90 degrees. Since we know from the key idea that the tangent at any point of a circle and the radius through this point are perpendicular to each other. Now from the figure we have angle OCT is equal to angle OCA plus angle ACT. Now OCT is of measure 90 degrees. Then this is equal to angle OCA plus angle ACT which is of measure 48 degrees. So this means that angle OCA is equal to 90 degrees minus 48 degrees. This means angle OCA is of measure 42 degrees. So we get this angle OCA of measure 42 degrees. Now consider the triangle ACT in this triangle. The exterior angle that is angle CAB is equal to angle ACT plus angle ATC. Since we know that in a triangle exterior angle is equal to the sum of interior opposite angles. Therefore we get that exterior angle CAB is equal to angle ACT which is of measure 48 degrees plus angle ATC which is of measure 36 degrees and this is equal to 84 degrees. Thus angle CAB is of measure 84 degrees. Next we consider the triangle OAC. In this triangle OA is equal to OC since they are the radii of the circle. So they are equal and therefore angle OAC would be equal to angle OCA and we know that angle OCA is equal to 42 degrees. So both these angles are of measure 42 degrees since we know that in a triangle angles opposite equal sides are equal. Thus we get this angle is also of measure 42 degrees. Now again from the figure we observe that angle CAB is equal to angle OAC plus angle OAB. This means that angle CAB is of measure 84 degrees. So we have 84 degrees is equal to angle OAC which is of measure 42 degrees plus angle OAB. So from here we have angle OAB is equal to 84 degrees minus 42 degrees which is equal to 42 degrees. Thus angle OAB is also of measure 42 degrees. So this angle is also of measure 42 degrees. Now again consider the triangle OAB. In this triangle OA would be equal to OB since they are the radii of the circle. So they are equal and therefore angle OAB would be equal to angle OBA and each would be equal to 42 degrees since we know that in a triangle angles opposite equal sides are equal. So again we have this angle is also of measure 42 degrees. Now let's again consider the triangle OAB. In this angle OAB plus angle OBA plus angle AOB is equal to 180 degrees by the anglesome property of a triangle. So this means that 42 degrees plus 42 degrees that is angle OAB and OBA are of measure 42 degrees plus angle AOB is equal to 180 degrees. So here we have angle AOB would be equal to 180 degrees minus 84 degrees that is 42 degrees plus 42 degrees is 84 degrees and so this gives us angle AOB is equal to 96 degrees and this is what we were supposed to find out. So finally we have the angle subtended by AB at the center of the circle is 96 degrees. So this is our final answer. This completes the session. Hope you have understood the solution of this question.