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Math Help - optimizing triangle w/coordinates

  1. #1
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    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...

    any help? please??
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  2. #2
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    I figured it out.. yay!
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  3. #3
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    Quote Originally Posted by littlejodo View Post
    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]?
    You do not need any Calculus to solve this problem.
    Just reflect through the x-axis.

    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
    Attached Thumbnails Attached Thumbnails optimizing triangle w/coordinates-conformal.jpg  
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