 Hello and welcome to the session. In this session we discussed the following question which says in triangle PQR, Pt is perpendicular to QR and Qt is equal to 3 times Tr proves that 2PQ square is equal to 2PR square plus QR square. Let's see the solution now. We are given this triangle PQR in which we have Pt is perpendicular to QR and we are also given that Qt is equal to 3 times Tr. We need to prove that 2PQ square is equal to 2PR square plus QR square. We have Qt is equal to 3 times Tr. Now from the figure we have that QR is equal to Qt plus Tr. Now as Qt is equal to 3 times Tr. So this means that QR is equal to 3 times Tr plus Tr. So we get QR is equal to 4 Tr. This means we have that Tr is equal to 1 upon 4 QR. We take this as equation 1. Now consider the triangle PQT in this angle PtQ is 90 degree. So this means triangle PQT is a right angle triangle. So therefore by the Pythagoras theorem we have PQ square is equal to Pt square plus TQ square. Let this be equation 2. Now consider the triangle PtR in this angle PtR is equal to 90 degrees. So it's a right angle triangle and so we apply the Pythagoras theorem in this triangle also. So by the Pythagoras theorem we have R square is equal to Pt square plus Tr square. Let this be equation 3. Now subtracting equation 3 from equation 2 we get PQ square minus Pr square is equal to Pt square plus TQ square minus Pt square minus Tr square. This means we get PQ square minus Pr square is equal to TQ square minus Tr square. Now since we have that QT is equal to 3 times Tr. So this means PQ square minus Pr square is equal to 3 times Tr the whole square minus Tr square since we know that QT or TQ is equal to 3 times Tr. This further gives us PQ square minus Pr square is equal to 9 Tr square minus Tr square. Further we get PQ square minus Pr square is equal to 8 Pr is equal to 1 upon 4 into QR where we get Q square minus Pr square is equal to 8 into 1 upon 4 into the whole square since we have Tr is equal to 1 upon 4 into QR. So this gives us PQ square plus square is equal to 8 into 1 upon 16 into QR square. Now 8 2 times is 16. So we get PQ square minus Pr square is equal to QR square or you can say TQ square minus 2 into Pr square is equal to QR square that is we have square is equal to 2 Pr square plus QR square and we will suppose to prove this. So hence through the C session hope you have understood the solution of this question.