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Question Video: Using the Angle Bisector Theorem to Find the Ratio between Two Unknown Side Lengths Mathematics • 11th Grade

In the shown figure, determine ๐ด๐ท : ๐ต๐ท.

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Video Transcript

In the shown figure, determine ๐ด๐ท to ๐ต๐ท.

In the diagram, we can observe that we have a larger triangle, triangle ๐ด๐ต๐ถ, with two given side lengths of nine centimeters and 16 centimeters. And we can observe that the angle at vertex ๐ถ has been bisected because these two angle portions are marked as congruent. We are asked to work out the ratio of ๐ด๐ท to ๐ต๐ท. 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 ๐ด๐ท and ๐ต๐ท have the same ratio as the side lengths of ๐ด๐ถ and ๐ต๐ถ. We can write this mathematically as ๐ด๐ท to ๐ต๐ท equals ๐ด๐ถ to ๐ต๐ถ.

As we are given that the side lengths of ๐ด๐ถ and ๐ต๐ถ are nine centimeters and 16 centimeters, then the ratio can be written as nine to 16. And so the ratio of ๐ด๐ท to ๐ต๐ท 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.

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