 Hello and welcome to the session. Let us understand the following question today. The perpendicular from A on side BC of a triangle intersects BC at D such that dB is equal to three times of CDC figure 6.55. Prove that twice of AB square is equal to twice of AC square plus BC square. Let us see the figure for the question. We have this triangle ABC such that dB is equal to thrice of CD. Now, given to us that dB is equal to thrice of CD, let us name this equation as number 1. Now, BC is equal to, it can be written as the sum of CD and dB. Now, BC is equal to CD plus dB can be replaced by thrice of CD using one. Now, BC is equal to three CD plus CD gives us four CD which can be written as 1 by 4 times of BC is equal to CD. Let us name this equation as number 2. Now, from 1 we have BD is equal to three times of CD. Therefore, BD is equal to three multiplied by CD is equal to 1 by 4 BC. Therefore, BD is equal to 3 by 4th of BC. Let us name it as number 3. Now, since triangle ABD is a right triangle ABD which is right angle at D. Therefore, by Pythagoras theorem we have AB square which is the hypotenuse is equal to AD square plus dB square. Let us name this equation as number 4. Similarly, in triangle ACD we have ACD AC square that is the hypotenuse is equal to the sum of the squares of the other two sides that is AD square plus CD square. Let us name it as number 5. Now, subtracting equation 5 from 4 we get AB square minus AC square is equal to AD square plus dB square minus AD square minus CD square. Now, here we see that AD square gets cancelled with minus AD square. So, we are left with AB square minus AC square is equal to dB square minus CD square. Now, which implies AB square minus AC square is equal to dB we can see is equal to 3 by 4 BC by 3. So, we can replace it by 3 by 4 BC the whole square minus CD is equal to 1 by 4 BC by 2. So, we can replace it by 1 by 4 BC the whole square using 2 and 3. Now, it implies AB square minus AC square is equal to 9 by 16 BC square minus 1 by 16 BC square which implies AB square minus AC square is equal to 8 by 16 BC square. Now, this gets cancelled by 2. So, we get AB square minus AC square is equal to 1 by 2 BC square which implies 2 AB square minus 2 AC square is equal to BC square which implies 2 AB square is equal to 2 AC square plus BC square. And this is our required result. Hence proved, I hope you understood the question. Bye and have a nice day.