Question Video: Finding the Ratio between Two Side Lengths Using the Angle Bisector Theorem Mathematics

In the given figure, if 𝐴𝐡 : 𝐴𝐢 : 𝐡𝐢 = 6 : 9 : 11, find the ratio of 𝐡𝐷 : 𝐷𝐢.

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Video Transcript

In the given figure, if the ratio 𝐴𝐡 to 𝐴𝐢 to 𝐡𝐢 equals the ratio six to nine to 11, find the ratio of 𝐡𝐷 to 𝐷𝐢.

On the figure, we can note down the given ratios. 𝐴𝐡 is six parts of that ratio, 𝐴𝐢 is nine parts of the ratio, and 𝐡𝐢 is 11 parts of the ratio. We are then asked to work out the ratio of the two lower line segments, 𝐡𝐷 and 𝐷𝐢. In order to do this, we’ll use the fact that the angle measure at 𝐢𝐴𝐡 has been bisected. We know this because the two angles 𝐢𝐴𝐷 and 𝐷𝐴𝐡 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 𝐷𝐢 and 𝐷𝐡 will be the same as the ratio of 𝐴𝐢 and 𝐴𝐡.

To write this mathematically, we could say that the ratio 𝐡𝐷 to 𝐷𝐢 is equal to 𝐴𝐡 to 𝐴𝐢. And we were given in the question that the ratio of 𝐴𝐡 to 𝐴𝐢 is equal to six to nine. Therefore, 𝐡𝐷 to 𝐷𝐢 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 𝐡𝐢 is simply a ratio part that we didn’t need in the question. But here we can give the answer that the ratio 𝐡𝐷 to 𝐷𝐢 is two to three.

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