What is 𝐴𝐷 squared equal to?
Some of you when you look at this triangle might immediately start thinking about algebra and substitution. And if you try to do this that way, you’re going to spiral down into a very long trail of substitution. There is a more simple way to approach this.
We’re going to set up a proportion. Here’s triangle 𝐴𝐵𝐷 and triangle 𝐴𝐷𝐶. Proportionally, 𝐵𝐷 over 𝐴𝐷 would be equal to 𝐴𝐷 over 𝐶𝐷. Once we have this proportion — 𝐵𝐷 over 𝐴𝐷 is equal to 𝐴𝐷 over 𝐶𝐷 — we can cross-multiply. 𝐴𝐷 times 𝐴𝐷 is equal to 𝐵𝐷 times 𝐶𝐷. 𝐴𝐷 times 𝐴𝐷 is equal to 𝐴𝐷 squared, which is equal to 𝐵𝐷 times 𝐶𝐷.