 Hi and welcome to the session my name is Shashi and I am going to help you with the following question. Question is in figure 6.59 abc is a triangle in which angle abc is less than 90 degree and ad is perpendicular to bc prove that ac square is equal to ab square plus bc square minus 2 bc multiplied by pd. Let us start the solution now first of all we will write whatever is given in the question we have given this figure and angle abc is less than 90 degree and ad is perpendicular to bc. So we can write in triangle abc ad is perpendicular to bc and angle abc is less than 90 degree. Now we will write what we have to prove so we will write to prove ac square is equal to ab square plus bc square minus 2 bc multiplied by pd. Let us start the proof now we know ad is perpendicular to bc so this implies triangle adb is right triangle right angle at d similarly triangle adc is right triangle right angle at d so first of all we will consider triangle adb so we can write in right triangle adb ad square is equal to ad square plus bd square by pythagoras theorem now let us prove that ad is perpendicular to bc. Now let us name this expression as 1 now we will consider that triangle adc so we will write in right triangle adc ac square is equal to ad square plus cd square this is my pythagoras theorem we know cd is equal to bc minus bd. So we can write this implies ac square is equal to ad square plus we will substitute for cd bc minus bd now this implies ac square is equal to ad square plus bc square plus bd square minus 2 bc multiplied by bd here we have applied a minus b whole square formula for this bracket we know a minus b whole square is equal to a square plus b square minus 2 ad so we have written bc square plus bd square minus 2 bc multiplied by bd now this implies ac square is equal to ad square plus bd square plus bc square minus 2 bc multiplied by bd we know ad square plus bd square is equal to ab square from expression 1 so we can write ac square is equal to ab square plus bc square minus 2 bc multiplied by bd plus we get ac square is equal to ad square plus bc square minus 2 bc multiplied by bd hence proved this completes the session hope you understood the session take care and goodbye