Question Video: Finding the Length of a Side in a Triangle Using the Right-Angled Triangle Altitude Theorem Mathematics

Video Transcript

Find the length of the line segment 𝐴𝐷.

To find the length 𝐴𝐷, we first note that 𝐷 is the projection of 𝐵 onto 𝐴𝐶 and that the triangle 𝐴𝐵𝐶 is a right triangle at 𝐵. And now recalling the Euclidean theorem, this tells us that 𝐴𝐵 squared is 𝐴𝐷 multiplied by 𝐴𝐶. That is, the side length 𝐴𝐵 squared is the product of the segment 𝐴𝐷 with the length of the hypotenuse 𝐴𝐶. We’re given that 𝐴𝐵 is 34 centimeters and that the hypotenuse 𝐴𝐶 is 40 centimeters. And substituting these values into the Euclidean theorem, this gives us 34 squared is 𝐴𝐷 multiplied by 40. Dividing through by 40 and rearranging, this gives us 𝐴𝐷 is 34 squared over 40, that is, 1156 divided by 40, which is 28.9. The line segment 𝐴𝐷 is therefore 28.9 centimeters.