‘And Jesus said to them, “Neither do I tell you by what authority I do these things.” '
So when Jesus then declared that He was not willing to submit His case to the very people who had admitted that they did not know how to judge a prophet's authority, the people would recognise that He had really answered their question. His claim was that the source of His authority was the same as that of John, which was what they thought anyway. The Sanhedrin were stymied, and the belief of the people was thus confirmed.