# Question Video: Finding the Ratio by Which a Point Divides a Line Segment Mathematics

If πΆ β π΄π΅ and vector π΄π΅ = 3 vector πΆπ΅, then πΆ divides vector π΅π΄ by the ratio οΌΏ. [A] 2 : 1 [B] 1 : 2 [C] 1 : 3 [D] 3 : 1

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### Video Transcript

If πΆ is an element of π΄π΅ and vector π΄π΅ equals three times vector πΆπ΅, then πΆ divides vector π΅π΄ by the ratio blank. Option (A) two to one, option (B) one to two, option (C) one to three, option (D) three to one.

Thereβs quite a lot of information in this question. But letβs start with the fact that we have this line segment π΄π΅, which we could model like this. Weβre told that πΆ is an element of π΄π΅, so that means that there will be a point πΆ somewhere on this line. If vector π΄π΅ is equal to three times vector πΆπ΅, then that means that three of this length πΆπ΅ would make up the length of π΄π΅. We could divide our length π΄π΅ into three pieces, but the question is, is πΆ here or is πΆ here? If we consider if πΆ is at this lower point, then the length πΆπ΅ would look like this. But if we were to multiply πΆπ΅ by three, we wouldnβt get the length of π΄π΅. We can then say that πΆ must be here, closer to π΅, as this length of πΆπ΅ would fit. Three lots of πΆπ΅ would give us π΄π΅.

We now need to work out the question of how πΆ divides this vector π΅π΄. We can then say if π΅πΆ is one unit length long, then π΄πΆ would be equivalent to two of these lengths. So do we write this ratio as two to one or one to two? Well, the direction here is very important. Weβre given the vector π΅π΄, so that means that weβre going from π΅ to π΄. We can therefore give our answer that itβs the ratio one to two, which is given in option (B). Note that if we had been given the vector π΄π΅ instead, then that wouldβve been the ratio two to one. But here, since πΆ is dividing vector π΅π΄, then itβs the ratio one to two.