This is what I came up with, though this might not be the intended approach!
I found and labeled the y-intercept and x-intercept of line A, then used that to label the lengths of the blue and pink lines. I labeled the green line "1s" and the orange line "2s" because the slope of line A is 2. So, for every 1 unit line A goes right, it goes up 2 units. " s " is the unknown scale factor.
the x-coordinate of P = blue line + green line
Let's just look at this right triangle:
We can use the Pythagorean theorem to solve for s.
\((\frac52+1s)^2+(2s)^2=10^2 \\ \frac{25}{4}+5s+5s^2=100 \\ s^2+s+\frac14=20-\frac54+\frac14 \\ (s+\frac12)^2=19 \\ s=\pm\sqrt{19}-\frac12 \\ s=\sqrt{19}-\frac12\)
So...
the x-coordinate of P \(=\frac52+\sqrt{19}-\frac12 \\~\\ = 2+\sqrt{19}\)
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