In the shown figure, determine 𝐴𝐷 to 𝐵𝐷.
In the diagram, we can observe that we have a larger triangle, triangle 𝐴𝐵𝐶, with two given side lengths of nine centimeters and 16 centimeters. And we can observe that the angle at vertex 𝐶 has been bisected because these two angle portions are marked as congruent. We are asked to work out the ratio of 𝐴𝐷 to 𝐵𝐷. Observe that this is a ratio and not the actual side length.
In order to do this, we can apply one of the angle bisector theorems. And because the bisected angle is inside the triangle, then we will 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. Here, this means that the line segments on the opposite side of 𝐴𝐷 and 𝐵𝐷 have the same ratio as the side lengths of 𝐴𝐶 and 𝐵𝐶. We can write this mathematically as 𝐴𝐷 to 𝐵𝐷 equals 𝐴𝐶 to 𝐵𝐶.
As we are given that the side lengths of 𝐴𝐶 and 𝐵𝐶 are nine centimeters and 16 centimeters, then the ratio can be written as nine to 16. And so the ratio of 𝐴𝐷 to 𝐵𝐷 is also nine to 16. As this ratio cannot be simplified any further, then we can give the answer that the ratio is nine to 16.