[Math] How to find the coordinates where the altitude of a triangle intersects the base

Question

I have a triangle with known, but random coordinates for each point.

Let's assume A(3,4), B(5,7), C(13.5,8.5)

How can I find the coordinates where the altitude from point B intersects the AC segment?

I've calculated the length of BD, AD and DC segments, the slope of AC but I'm not sure what's next.

Answer 1

The line $AC$ is defined by $$\frac{y-y_A}{x-x_A}=\frac{y_C-y_A}{x_C-x_A}$$

Then we know $AC(x)=m_{AC}x+b_{AC}$, where $m_{AC}$ is the slope and $b_{AC}$ is the y-intercept (both are known).

Let $BD(x)=m_{BD}x+b_{BD}$. Then $m_{BD}=\frac{-1}{m_{AC}}$ and $b_{BD}=y_B-m_{BD}x_B$.

Now solve $AC(x)=BD(x)$.