 Hello and welcome to the session. Let's work out the following problem. It says in a right angle to triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides Use the above to prove the following in a right in a triangle ABC AD is perpendicular on BC Prove that AB squared plus CD squared is equal to AC squared plus DD squared. Let's now move on to the solution. We'll first prove the first part Which says that in a right angle triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides This is called Pythagoras theorem. So let's first write what is given to us We are given a right triangle ABC right angle at B. Let's now write what we have to prove Now since The triangle is right angle at B. AC is the hypotenuse We have to prove that the square of the hypotenuse is equal to the sum of the squares of the other two sides So AC squared is equal to the AB squared plus BC squared. This is what we have to prove and to prove this we do the following construction draw Vendicula BD on AC Right, let's now start the proof now in triangle ADB and ABC Angle A is equal to angle A since it's a common angle angle ADB is equal to angle C DB It's AD It's angle ABC Because each is 90 degrees right so this implies Triangle ADB is similar to triangle ABC By a similarity Now since triangle ADB is similar to triangle ABC The ratio of the corresponding sides will be equal that is AD upon AB is equal to AB upon AC Now similarly in triangle CDB Triangle CBA Again angle C is equal to angle C and Angle CDB is equal to angle CBA Common angle and this is each is 90 degrees So this implies triangle CDB is Similar to triangle CBA So again ratio of the corresponding sides will be same so CD upon CB is equal to CB upon CA so this implies BC square is equal to CD into CA and from this we have AB square is equal to AD into AC Let's name this as one and this as two adding one and two we have AB square plus AC square is Equal to AD into AC plus CD into AC Taking AC common we have AD plus CD now AD plus CD is AC So we have AC into AC AC into AC is AC square So we have square of the hypotenuse is equal to Some of the squares of the other two sides and here we have BC square So the first part is proved Now in the second part we have to prove that in a triangle ABC AD is a perpendicular on BC We have to prove that AB square plus CD square is equal to AC square plus BD square So let's now Solve the second part now in right triangle ADB AB square is equal to AD square plus BD square Because AB is the hypotenuse of the right triangle ADB so this implies AD square is equal to AB square minus VD square now in right triangle ADC AC square is equal to AD square plus CD square So this implies AD square is equal to AC square minus CD square Let's name this as one and this as two now let this be Name as one and this as two now equating one and two since both are values of AD square we have AB square minus VD square is equal to AC square minus CD square. So this implies AB square plus CD square is equal to AC square plus VD square and this is What we have to prove But this completes the question and the session life for now take care. Have a good day