 Hello and welcome to the session. In this session we discuss the following question that says, in the right triangle ABC, the perpendicular BD on the hypotenuse AC is drawn for that, first AC into AD is equal to AB square, second AC into CD is equal to BC square. Before we move on to the solution, let's discuss some results to be used in this question that says the tangent, any point of a circle and the radius through this point are perpendicular to each other. The next result is if a chord intersects internally, then the product of this segment is square at the tangent from the point of contact to the point of intersection. This is the key idea that we use for this question. Let's proceed with the solution now. This figure we are given that triangle ABC is the right triangle and also we have that BD is perpendicular to AC. First we need to prove that AC into AD is equal to AB square. We have this circle with BC as the diameter that is BC is the diameter circle and this angle ABC is equal to 90 degrees so this ABC is the right angle triangle therefore is the tangent to the circle at the point B. Since we know that the tangent at any point of a circle and the radius through this point are perpendicular to each other, therefore we can say that since BC is the diameter and this angle is of measure 90 degrees, therefore this AD is the tangent to the circle at the point B. Also CD is the chord of the circle. We have tint AB and the chord CD intersect each other externally at point A and so the key idea we have is if a chord and a tangent intersect externally then the product of the length of the segment of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection. So therefore product of the segment of the chord CD is given by AC into AD is equal to the square of the length of the tangent which is AB square. So AC into AD is equal to AB square and this is what we were supposed to prove. So here is proved AC into AD is equal to AB square. Now in the second part we need to prove AC into CD is equal to BC square. Now consider AC into CD this is equal to AC into, now from the figure we have CD is equal to AC minus AD this is equal to AC square minus AC into AD. Now that we get this is equal to AC square minus AD square since we have already proved that AC into AD is equal to AD. Now consider the right triangle ADC in this AC square is equal to AD square plus DC square by the Pythagoras and from here we have AC square minus AD square is equal to DC square. Therefore AC into CD is equal to AC square minus AB square which is DC square and this is where we were supposed to prove that AC into CD is equal to DC square but AC into CD is equal to DC square. This can be easy session hope you have understood the solution of this question and this is where we were supposed to prove that AC into CD is equal to DC square. Now consider that AC into CD is equal to DC square minus AC into CD is equal to AC square minus AC into CD is equal to DC square. This can be easy session hope you have understood the solution of this question.