In the given figure, if the ratio 𝐴𝐵 to 𝐴𝐶 to 𝐵𝐶 equals the ratio six to nine to 11, find the ratio of 𝐵𝐷 to 𝐷𝐶.
On the figure, we can note down the given ratios. 𝐴𝐵 is six parts of that ratio, 𝐴𝐶 is nine parts of the ratio, and 𝐵𝐶 is 11 parts of the ratio. We are then asked to work out the ratio of the two lower line segments, 𝐵𝐷 and 𝐷𝐶. In order to do this, we’ll use the fact that the angle measure at 𝐶𝐴𝐵 has been bisected. We know this because the two angles 𝐶𝐴𝐷 and 𝐷𝐴𝐵 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 𝐷𝐶 and 𝐷𝐵 will be the same as the ratio of 𝐴𝐶 and 𝐴𝐵.
To write this mathematically, we could say that the ratio 𝐵𝐷 to 𝐷𝐶 is equal to 𝐴𝐵 to 𝐴𝐶. And we were given in the question that the ratio of 𝐴𝐵 to 𝐴𝐶 is equal to six to nine. Therefore, 𝐵𝐷 to 𝐷𝐶 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 doesn’t 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 𝐵𝐶 is simply a ratio part that we didn’t need in the question. But here we can give the answer that the ratio 𝐵𝐷 to 𝐷𝐶 is two to three.