Question Video: Using the Angle Bisector Theorem to Find the Ratio between Two Unknown Side Lengths Mathematics • First Year of Secondary School

Video Transcript

In the figure, the ratio 𝐴𝐵 to 𝐴𝐶 equals four to seven. What is the ratio of 𝐵𝐷 to 𝐵𝐶?

In the figure, we can observe that this triangle 𝐴𝐵𝐶 has had its exterior angle bisected. We know this because the two angles are marked as congruent. We can therefore apply the exterior angle bisector theorem, which gives us a relationship between the line segments in the figure. We know that 𝐵𝐷 over 𝐷𝐶 must be equal to 𝐴𝐵 over 𝐴𝐶. We could also write this in the form of a ratio as 𝐵𝐷 to 𝐷𝐶 equals 𝐴𝐵 to 𝐴𝐶.

We were given in the question that the ratio of 𝐴𝐵 to 𝐴𝐶 is equal to the ratio of four to seven. So, that means that the ratio of 𝐵𝐷 to 𝐷𝐶 is also four to seven. But we must be careful because this is not the ratio that we were asked for in the question, which is 𝐵𝐷 to 𝐵𝐶. So, we need to find the ratio of this segment 𝐵𝐷 to this segment 𝐵𝐶. As we know that 𝐵𝐷 is four parts of the total of seven ratio parts, then the line segment of 𝐵𝐶 must make up three of the parts of the ratio. Therefore, we can write the ratio of 𝐵𝐷 to 𝐵𝐶 as four to three. And so we have answered the question by using the exterior angle bisector theorem.