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An event is shown as a point on a spacetime diagram

Another frame has coordinates of space and time as:

##\begin{aligned} c t^{\prime} &=(\cosh \theta)(c t)-(\sinh \theta) x \\ x^{\prime} &=(-\sinh \theta)(c t)+(\cosh \theta) x \\ y^{\prime} &=y \\ z^{\prime} &=z \end{aligned}##

It's said that the spacetime diagram corresponding to the new frame could be drawn on the previous spacetime diagram as shown below:

Then is that the time coordinate of the event in the new frame can be found as shown below (red line is parallel to the ##x′## line, ##(a,b)## is the intersection point of red line with ##ct′##axis)

Hence the new time coordinate is

##c t^{\prime}=\sqrt{a^2+b^2}##

*?*Is that correct?