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Question Video: Using the Angle Bisector Theorem to Find an Unknown Term Mathematics • 11th Grade

Given that 𝐴𝐢 = 10, 𝐢𝐷 = 6, 𝐴𝐡 = π‘₯ + 9, and 𝐡𝐷 = π‘₯ + 5, find the numerical value of π‘₯.


Video Transcript

Given that 𝐴𝐢 equals 10, 𝐢𝐷 equals six, 𝐴𝐡 equals π‘₯ plus nine, and 𝐡𝐷 equals π‘₯ plus five, find the numerical value of π‘₯.

We can begin this question by adding on the length information that we were given. 𝐴𝐢 is 10, 𝐢𝐷 is six, 𝐴𝐡 is equal to π‘₯ plus nine, and 𝐡𝐷 is equal to π‘₯ plus five. In this question, we need to work out this value of π‘₯, which appears in two of the side lengths. In order to do this, we can use the fact that angle 𝐢𝐴𝐡 is bisected. This is because we are given on the diagram that the measure of angle 𝐢𝐴𝐷 is equal to the measure of angle 𝐡𝐴𝐷.

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 𝐷𝐢 and 𝐷𝐡 have the same ratio as the two side lengths 𝐴𝐢 and 𝐴𝐡. We can write this mathematically as 𝐢𝐷 over 𝐡𝐷 equals 𝐴𝐢 over 𝐴𝐡.

Now, all we need to do is fill in the given side lengths and solve for π‘₯. So we have six over π‘₯ plus five is equal to 10 over π‘₯ plus nine. We can then cross multiply so that we have six times π‘₯ plus nine equals 10 times π‘₯ plus five. We can then distribute the six and the 10 across the parentheses so that we have six π‘₯ plus 54 is equal to 10π‘₯ plus 50. We then need to collect like terms. And since we have a larger value of 10π‘₯ on the right-hand side, then we can start by subtracting six π‘₯ from both sides. This simplifies to 54 is equal to four π‘₯ plus 50. We can then subtract 50 from both sides, which gives four is equal to four π‘₯.

Finally, we can divide through by four such that one is equal to π‘₯ or π‘₯ is equal to one. And it’s always worthwhile checking our values. In this case, as π‘₯ is equal to one, the side length of 𝐴𝐡 would be 10. The side length of 𝐡𝐷 would be six. In this specific example, the two side lengths of 𝐴𝐡 and 𝐴𝐢 were congruent and so were the side lengths of 𝐢𝐷 and 𝐷𝐡. This is not always the case with the angle bisector theorem. However, as our proportion statement does apply, then the answer of π‘₯ equals one must be correct.

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