1. ## optimizing triangle w/coordinates

Let Q = (0,7) and R=(9,12) be given points in the plane. Find the point P = (x, 0) so that the sum of distances PQ + PR is as small as possible. To do this what function of x do we need to minimize over the closed interval [0,9]?

I've drawn this out the best I can and I can see that it is all about triangles. Using the Pythagorean theorem I can use the x/y axis to make one triangle and I can create another using the x axis and y = 12. This way I need to find an x value such that:

x^2 + 7^2 = ? and (9-x)^2 + 12^2 = ? so that the sum of the two ?s is minimized.

I feel like this is a big accomplishment for me given my past troubles in understanding word problems, but I am still unsure how to relate these two in a single function to minimize...

Here are two illustrations below. Notice the $PR + QR = PR + Q'P$ because of the reflection we did. In the first picture the point $P$ is a little off so that the red and blue line are not straight when combined together. While in the second picture the point $P$ is in such a position so that the the blue line and red line coincide. The distance in the second picture is definitely smaller. Since the shortest distance between the two point is a straight line it means $P$ must be the intersection of $Q'P$ with the x-axis. The equation of $Q'P$ is $y + 7 = \tfrac{19}{9}x$ it intersections when $y=0 \implies x = \tfrac{63}{19}\approx 3.31$