 Hello and welcome to the session. I am Deepika and I am going to help you to solve the following question. The question says P and Q are points on sides C A and C B respectively of triangle A, B, C right angled at C. Prove that AQ square plus BP square is equal to AB square plus BQ square. Now we know that according to Pythagoras theorem in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. So this is a key idea behind our question. We will use this theorem to solve the above question. So let's start the solution. Now we are given P and Q are points on sides C A and C B respectively of triangle ABC which is right angled at C. So we are given P and Q are points on sides C A and C B respectively of triangle ABC which is right angled at C. We have to prove that AQ square plus BP square is equal to AB square plus BQ square. So we can write we have to prove AQ square plus BP square is equal to AB square plus PQ square. So in this figure, let us join AQ, BP and PQ. Now in triangle ABC right angled at C we have AB square is equal to AC square plus PC square because according to the Pythagoras theorem in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. So let us give this as number one. Again in right triangle AQC we have AQ square is equal to AC square plus QC square. Let us give this as number two and here also we have used the Pythagoras theorem. Again in right triangle AQC which is right angled at C we have PQ square is equal to PC square plus QC square. Let us give this as number three. Again here we have used the Pythagoras theorem. Again in right triangle PBC we have PBC is equal to PC square plus BC square. Let us give this as number four. Now we have to prove that AQ square plus BP square is equal to AB square plus PQ square. Now from two we have AQ square is equal to AC square plus QC square and from four we have PBC square is equal to PC square plus BC square. So on adding two and four we get AQ square plus BP square is equal to AC square plus QC square plus PC square plus BC square or this can be written as AC square plus BC square plus QC square plus PC square. Now from one we have AB square is equal to AC square plus PC square and from three we have PQ square is equal to PC square plus QC square. So by using one and three we have AQ square plus BP square is equal to AB square plus PQ square. Therefore AQ square plus BP square is equal to AB square plus PQ square. Hence we have proved that required result. This completes our session. I hope the solution is clear to you. Bye and have a nice day.