Video Transcript
Using the figure below, find the length of line segment 𝐴𝐷 to the nearest hundredth.
In the figure below, we have a triangle 𝐴𝐵𝐶 with two given line segment lengths of 27 centimeters and 29 centimeters. We can also observe that angle 𝐴𝐵𝐶 is a right angle. And it’s being bisected because angle 𝐶𝐵𝐷 is marked as congruent with angle 𝐷𝐵𝐴. The fact that this interior angle 𝐴𝐵𝐶 has been bisected will allow us to apply 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. By this theorem, we can then write that 𝐴𝐷 over 𝐶𝐷 is equal to 𝐴𝐵 over 𝐶𝐵. As we are given that the lengths of 𝐴𝐵 and 𝐶𝐵 are 27 and 29 centimeters, respectively, then they are in the ratio of 27 to 29. And so 𝐴𝐷 to 𝐶𝐷 will also have the ratio of 27 to 29.
The important thing to remember when we are applying this angle bisector theorem is that we are not saying that the lengths are congruent. We are not saying that 𝐴𝐷 and 𝐶𝐷 are 27 centimeters and 29 centimeters. Remember, this is just giving us a ratio. However, we do know that the line segment 𝐴𝐶 is split into this ratio. And we can use the fact that 𝐴𝐵𝐶 is a right triangle to help us work out the length of the line segment 𝐴𝐶.
We recall that the Pythagorean theorem tells us that in any right triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides. In triangle 𝐴𝐵𝐶, the line segment 𝐴𝐶 represents the hypotenuse. The two shorter sides are 27 centimeters and 29 centimeters. Therefore, substituting these in to the Pythagorean theorem, we have 27 squared plus 29 squared equals 𝐴𝐶 squared.
We could then evaluate the squares on the left-hand side, giving us 729 plus 841, which simplifies to 1570. We can then take the square root of both sides of this equation, which gives us that 𝐴𝐶 is equal to the square root of 1570 centimeters. Now, because we haven’t quite finished with this calculation, we want to keep our answer for 𝐴𝐶 as accurate as possible. Therefore, we will use the square root of 1570. But if we’re changing it into a decimal, it would be approximately 39.6232 and so on centimeters. If we do change it to a decimal, we want to keep this nonrounded value in the calculator rather than rounding it too early and getting a more inaccurate answer.
So, so far, we’ve worked out that the line segment 𝐴𝐶 is the square root of 1570 centimeters. But we know that the ratio between the line segments 𝐴𝐷 and 𝐶𝐷 is 27 to 29. We want to find the actual length of the line segment 𝐴𝐷. This question has then become a ratio problem. So, in order to calculate the value of 𝐴𝐷, we would add the ratio parts. 27 plus 29 is equal to 56. So 𝐴𝐷 is equal to 27 over 56 multiplied by the length of 𝐴𝐶, which is the square root of 1570. Typing this into our calculators, we would get a decimal value of 19.104 and so on centimeters. Rounding this to the nearest hundredth as required in the question, we have the answer that the length of 𝐴𝐷 is 19.10 centimeters to the nearest hundredth.
