Given that 𝐴𝐶 equals 10, 𝐶𝐷 equals six, 𝐴𝐵 equals 𝑥 plus nine, and 𝐵𝐷 equals 𝑥 plus five, find the numerical value of 𝑥.
We can begin this question by adding on the length information that we were given. 𝐴𝐶 is 10, 𝐶𝐷 is six, 𝐴𝐵 is equal to 𝑥 plus nine, and 𝐵𝐷 is equal to 𝑥 plus five. In this question, we need to work out this value of 𝑥, which appears in two of the side lengths. In order to do this, we can use the fact that angle 𝐶𝐴𝐵 is bisected. This is because we are given on the diagram that the measure of angle 𝐶𝐴𝐷 is equal to the measure of angle 𝐵𝐴𝐷.
We recall 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 respective bisected angle. This does sound rather complicated. But it’s really telling us that these two line segments of 𝐷𝐶 and 𝐷𝐵 have the same ratio as the two side lengths 𝐴𝐶 and 𝐴𝐵. We can write this mathematically as 𝐶𝐷 over 𝐵𝐷 equals 𝐴𝐶 over 𝐴𝐵.
Now, all we need to do is fill in the given side lengths and solve for 𝑥. So we have six over 𝑥 plus five is equal to 10 over 𝑥 plus nine. We can then cross multiply so that we have six times 𝑥 plus nine equals 10 times 𝑥 plus five. We can then distribute the six and the 10 across the parentheses so that we have six 𝑥 plus 54 is equal to 10𝑥 plus 50. We then need to collect like terms. And since we have a larger value of 10𝑥 on the right-hand side, then we can start by subtracting six 𝑥 from both sides. This simplifies to 54 is equal to four 𝑥 plus 50. We can then subtract 50 from both sides, which gives four is equal to four 𝑥.
Finally, we can divide through by four such that one is equal to 𝑥 or 𝑥 is equal to one. And it’s always worthwhile checking our values. In this case, as 𝑥 is equal to one, the side length of 𝐴𝐵 would be 10. The side length of 𝐵𝐷 would be six. In this specific example, the two side lengths of 𝐴𝐵 and 𝐴𝐶 were congruent and so were the side lengths of 𝐶𝐷 and 𝐷𝐵. This is not always the case with the angle bisector theorem. However, as our proportion statement does apply, then the answer of 𝑥 equals one must be correct.