 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says the perpendicular from A on the side BC of a triangle ABC intersects BC at D such that dB is equal to 3 times CD. Prove that twice of AB square is equal to 2 into AC square plus BC square. So first of all let us write down what is given to us in the question. We are given that AD is perpendicular on BC and dB is equal to 3 times CD. We have to prove that 2 AB square is equal to 2 AC square plus BC square. So let us start with the proof to this question. First of all we see that in right angle triangle ADB, AB square will be equal to AD square plus BD square using the Pythagoras theorem. So we have in right angle triangle ADB, AB square is equal to AD square plus BD square and this we call 1. Now we see that BD is equal to BC minus DC. So BD is equal to BC minus CD. So we simply put the value of BD here and we get AB square is equal to AD square plus BC minus CD the whole square and that is equal to AD square plus BC square plus CD square minus 2 into BC into CD. Now we see that in right angle triangle ADC, AC square is equal to AD square plus DC square again using the Pythagoras theorem. So in right angle triangle ADC, we have AC square is equal to AD square plus CD square. So in the place of AD square plus CD square we can write AC square. We get AB square is equal to AC square plus BC square minus 2 into BC into CD. Now we have BD is equal to 3 times of CD that is given to us. We call this 2. Now here we add CD on both the sides. So we have BD plus CD is equal to 3 CD plus CD. This we get on adding CD on both sides. By doing so we see that BD plus CD is BC. So we write BC is equal to 4 times of CD or CD is equal to VC by 4. So now we substitute VC by 4 in place of CD in equation 2 that means this equation. So we will have now AB square is equal to AC square plus BC square minus 2 BC into BC by 4. Now we see that since 4 is twice of 2, so this becomes 2 here and we have this to be equal to AC square plus BC square minus BC square by 2. Now we multiply both sides by 2. So we get twice of AB square is equal to twice of AC square plus twice of BC square minus BC square or twice of AB square is equal to twice of AC square plus BC square. Hence proved. So I hope that you understood the question and enjoyed the session. Have a good day.