 Hello and welcome to the session. In this session we discussed the following question which says in the given figure pqrs is a quadrilateral in which angle s is equal to 90 degrees, a circle c with center o and radius r touches the sides pqqrrs and sp at p dash q dash r dash s dash respectively. If qr is equal to 10 centimeters, rs is equal to 20 centimeters and p dash q is equal to 5 centimeters, find the value of r. Before moving on to the solution let's recall one fact which says that the lengths of tangents drawn from an external point to a circle are equal. This is the key idea that we use in this question. Now we proceed with the solution. This is the figure given to us. Now let's see what all do we have here. We are given that pqrs is a quadrilateral and this angle s is equal to 90 degrees. Then we have qr is equal to 10 centimeters, rs is equal to 20 centimeters and p dash q is equal to 5 centimeters and we need to find the value of r. Now we are given a circle with center o and radius r. So we have to find the value of r that is we have to find the radius of the circle. First of all we join o r dash and o s dash. We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact. So this would mean that o r dash is perpendicular to s r dash then o s dash is perpendicular to s s dash. Now consider the figure o s dash s r dash in this we have all angles r of measure 90 degrees and also o s dash is equal to o r dash since they are the right of the same circle. Therefore we can say that o s dash s r dash is a parallelogram and so it means all sides are equal. That is we have o s dash is equal to s dash s is equal to s r dash is equal to r dash o and this implies that the figure o s dash s r dash is a square. Now p dash q is equal to q q dash since they are the tangents drawn from the point q so they are equal. Now p dash q is equal to 5 centimeters so this implies that q q dash is equal to 5 centimeters and from the figure q q dash is equal to r q minus r q dash. This gives us r q dash is equal to r q minus q q dash that is r q dash is equal to now r q is of measure 10 centimeters as it is given to us minus q q dash which is of measure 5 centimeters. So 10 centimeters minus 5 centimeters is equal to 5 centimeters. That is we have r q dash is equal to 5 centimeters. Now again r q dash is equal to r r dash since they are the tangents from the external point r and now r q dash is equal to 5 centimeters so this means that r r dash is also equal to 5 centimeters. Now from the figure we have r r dash is equal to s r minus s r dash or you can say from here we get s r dash is equal to s r or r s minus r r dash is given to us that s r or r s is of measure 20 centimeters so s r dash is equal to 20 centimeters minus r r dash which is 5 centimeters and this is equal to 15 centimeters so we get s r dash is equal to 15 centimeters. Now that we had obtained that o s dash s r dash is a square so this means o s dash is equal to s dash s is equal to s r dash is equal to r dash o that is all sides are equal. We have obtained s r dash is equal to 15 centimeters therefore we get o r dash is equal to o s dash is equal to 15 centimeters that is they are the delay of the circle so we get r is equal to 15 centimeters this is our final answer. This completes the session hope you have understood the solution of this question.