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The Minimum Value of The Sum

Hello everyone I have a question asking the minimum value for the sum of (|AP|+|PD|)How can I solve this can you explain step by step? Thanks for your contribution. *Please have a look at the attachment.*

PS:I have found the solution by writing two seperate cosinus theorem functions adding them together than taking the derivative of the resulting function, equaling the derivative function to "0" and substituting the root with x in the main function that gives the sum of |AP|+|PD| with the help of TI-89 Titanium but this exam must be solved without the aid of any kind of calculator. The Correct answer is