 Hi and welcome to the session. My name is Shashi. Let us discuss the following question. Question is in figure 6.58 ABC is a triangle in which angle ABC is greater than 90 degree and A day is perpendicular to C B. Pro fact AC square is equal to AB square plus BC square plus 2 multiplied by BC multiplied by BD. This is the figure 6.58. Let us now start with the solution. First of all we will write whatever is given in the question we are given a triangle ABC in which angle ABC is greater than 90 degree and AD is perpendicular to we have to prove AC square is equal to AB square plus BC square plus 2 BC BD. Let us start the proof now. Now we can clearly see that angle D is equal to 90 degree means AD is perpendicular to BC. AD is perpendicular to BC implies AD is perpendicular to BC. Since DC is the extension of BC only so AD is perpendicular to BC and angle D is equal to 90 degree. So we can write angle D is equal to 90 degrees since AD is perpendicular to BC. Now angle D is equal to 90 degree implies triangle ADB is a right triangle and triangle ADC is also a right triangle. Now first of all let us consider right triangle ADB so we will write in right triangle B, AD square is equal to AD square plus BD square by Pythagoras theorem this expression as 1. Now let us consider the triangle ADC we will write in right triangle ADC we get AC square is equal to AD square plus CD square clearly we can see AC is the hard part in use and AD is the perpendicular DC is the base. So AC square is equal to AD square plus DC square again by Pythagoras theorem you can see CD is equal to BD plus CB so we can write AC square is equal to AD square plus BD plus BC whole square this is because CD is equal to BD plus BC so we have substituted for CD BD plus BC. Now applying the formula for A plus B whole square we get AC square is equal to AD square plus BD square plus BC square plus 2 BD multiplied by BC below A plus B whole square is equal to A square plus B square plus 2 AB so we have applied the formula of A plus B whole square in this bracket. Now we know AD square plus BD square is equal to AB square so substituting for AD square plus BD square AB square in this expression we get AC square is equal to AB square plus BC square plus 2 BD multiplied by BC. So finally we get AC square is equal to AB square plus BC square plus 2 BD multiplied by BC or we can write it as AC square is equal to AB square plus BC square plus 2 BC multiplied by BC so this is our required answer hence this completes the session hope you understood the session goodbye