Question Video: Finding an Unknown Side Length in a Triangle Using the Angle Bisector Theorem Mathematics

Video Transcript

In the triangle 𝐴𝐵𝐶, 𝐴𝐵 is 76 centimeters, 𝐴𝐶 is 57 centimeters, and 𝐵𝐷 is 52 centimeters. Given that the line segment 𝐴𝐷 bisects the angle 𝐴 and intersects the line segment 𝐵𝐶 at 𝐷, determine the length of the line segment 𝐴𝐷.

We recall that if the line segment 𝐴𝐷 bisects the angle 𝐴 in a triangle, then we have the theorem 𝐴𝐷 is equal to the square root of 𝐴𝐶 multiplied by 𝐴𝐵 minus 𝐷𝐶 multiplied by 𝐷𝐵. Now from the question, we know that 𝐴𝐶 is equal to 57, 𝐵𝐷, which is 𝐷𝐵, is equal to 52, and that 𝐴𝐵 has length 76. And so to find the length 𝐴𝐷, we must first find the length 𝐷𝐶. To do this, we’re going to use the interior angle bisector theorem. This says that the bisector of an interior angle of a triangle divides the opposite side into segments whose lengths have the same ratio as the lengths of the noncommon adjacent sides of the bisected angle.

What this means in our triangle is that 𝐷𝐶 over 𝐵𝐷 is the same as 𝐴𝐶 over 𝐴𝐵. Now we know that 𝐵𝐷 is 52, 𝐴𝐶 is 57, and 𝐴𝐵 is 76. So we have 𝐷𝐶 over 52 is equal to 57 over 76. Now, multiplying both sides by 52, we’re left with 𝐷𝐶 on the left-hand side. And 57 divided by 76 multiplied by 52 is 39. So 𝐷𝐶 is 39 centimeters. We now have all the information we need to calculate 𝐴𝐷. Substituting in our values, we have 𝐴𝐷 is the square root of 57 times 76 minus 39 times 52, that is, the square root of 4332 minus 2028, which is the square root of 2304, that is, 48. The length of the line segment 𝐴𝐷 is therefore 48 centimeters.