 Hello and welcome to the session. In this session we will discuss the question which says that proof of the lengths of the tangents drawn from an external point to a circle are equal. Here is the above result in the following. A circle is inscribed in a triangle ABC touching AB, BC and AC, HP, Q and R respectively as shown in the figure that is this figure. If AB is equal to 10 cm, AR is equal to 7 cm and RC is equal to 5 cm then find the length of BC. Now before starting the solution of this question we should know our result. And that is the tangent is perpendicular to the radius through the point of contact. Now this result will work out as a key idea for solving out and now we will start with the solution. First of all we have to prove that the lengths of the tangents drawn from an external point to a circle are equal. Now given PQ and PR are the tangents from an external point to prove Q is equal to PR. Now here construction will be join OQ, OR, now we will join OR, OQ and OP. Now angle OQP is equal to 90 degrees and angle ORP is equal to 90 degrees. This is by using the result which is given in the key idea that is tangent is perpendicular to radius through the point of contact. So these angles are of 90 degrees. Now we will start with the proof. Here we will consider the triangle OQP and triangle OPR. So in triangle OQP and triangle OPR OQ is equal to OR since these are radii of the same circle. So OQ is equal to OR because these are radii of the same circle. Now let us name this as angle 1 and this as angle 2 and angle 1 is equal to angle 2 both of them are of 90 degrees. Angle 1 is equal to angle 2 each of 90 degrees and OP is equal to OP because it is common in both the triangles. So now OP is equal to OP because it is common. So by RHS property triangle OQP is congruent to triangle OPR. This implies PQ is equal to PR by CPCDC as of congruent triangles are congruent. So as these two triangles are congruent so we have PQ is equal to PR. It is proved that the tangent drawn from the external point to a circle are equal in length. Now we will use this result in solving this question. Now in this part given in this diagram AB is equal to 10 centimeters OR is equal to 7 centimeters OR is equal to 5 centimeters. Now as we have proved earlier so we know that tangent drawn from an external point to a circle equal in length. Now in this diagram from an external point A tangent's AP and AR are drawn to the circle therefore AR is equal to AP. This implies now here AR is 7 centimeters so therefore AP will be also equal to 7 centimeters. Now see the external point from where the tangent's are drawn to the circle. Therefore by the same result QC is equal to RC. Now RC is equal to 5 centimeters so therefore this implies QC is also 5 centimeters. Now in this question we have to find the length of DC. Now BP is equal to AB minus AP. Now AP is 7 centimeters and AB is 10 centimeters. So BP is equal to 10 centimeters minus 7 centimeters which is equal to 3 centimeters. Now here BU is the external point from where the tangent's BP and BQ are drawn to the circle. Now using the result which we have proved earlier that the tangent's drawn from the external point are equal in length. Now we have BQ is equal to BP. Now BP is equal to 3 centimeters so this implies BQ is also 3 centimeters. Now BC can be written as BQ so it is BQ plus QC. Now BQ is equal to 3 centimeters is equal to 5 centimeters. So this is equal to 3 centimeters plus 5 centimeters which is equal to 8 centimeters. Therefore length of the side BC is equal to 8 centimeters. That's the solution of the given question and that's all for this session. Enjoy the session.