Flash of light thought experiment, need explanation

marmot
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So suppose in frame K at x_1 a photon is emmited and its absorbed at x_2= x_1+L so, the photon in K has traveled a total distance of L. Now suppose we have frame K' moving at some velocity Beta*c. Why is it that in K' the distance traveled by the photon is different than L if the speed of light is constant? I did this problem and got that. I am scratching my head at this.
 
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Because in K', the distance the photon traveled and the time in which it traveled are shorted by the same multiplier- the Lorentz transform parameter.
 
I think that makes sense. So you need to fudge the time and distance interval so that the speed of light is constant, right?
 

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