### Video Transcript

In the given figure, if the ratio π΄π΅ to π΄πΆ to π΅πΆ equals the ratio six to nine to 11, find the ratio of π΅π· to π·πΆ.

On the figure, we can note down the given ratios. π΄π΅ is six parts of that ratio, π΄πΆ is nine parts of the ratio, and π΅πΆ is 11 parts of the ratio. We are then asked to work out the ratio of the two lower line segments, π΅π· and π·πΆ. In order to do this, weβll use the fact that the angle measure at πΆπ΄π΅ has been bisected. We know this because the two angles πΆπ΄π· and π·π΄π΅ are marked as congruent.

We can use the interior angle bisector theorem. This theorem states that if an interior angle of a triangle is bisected, the bisector divides the opposite side into segments whose lengths have the same ratio as the lengths of the noncommon adjacent sides of the bisected angle. It all sounds a little complicated, but all it really means is that the ratio of π·πΆ and π·π΅ will be the same as the ratio of π΄πΆ and π΄π΅.

To write this mathematically, we could say that the ratio π΅π· to π·πΆ is equal to π΄π΅ to π΄πΆ. And we were given in the question that the ratio of π΄π΅ to π΄πΆ is equal to six to nine. Therefore, π΅π· to π·πΆ is also equal to six to nine. And we can simplify this further to the ratio of two to three. And so we have found the answer.

The important thing in this question is to remember that the ratio of six to nine to 11 doesnβt represent length units. For example, here we have two ratios of nine to six, and they do not need to add up to 11. The value of 11 for this line segment of π΅πΆ is simply a ratio part that we didnβt need in the question. But here we can give the answer that the ratio π΅π· to π·πΆ is two to three.