Find the length of the red line

Question

I have been trying to solve this question from twitter for like two days could anyone help me with this?

All the credits go to the name mentioned in the image

My attempt to solve:

I think it's harder to understand my messy diagram. But all I have done here is drawing a parallel line from a vertice to the opposite side of the square and with that I have find the lengths of the mentioned segments in terms of x

Thank u in advance for looking into this problem

Answer 1

Drop a perpendicular from the center $O$ of the square to the bottom of the square. This creates a right triangle whose hypotenuse is the red segment and which is similar to the right triangle whose hypotenuse is the blue segment. If the square has side length $2x$, and $r$ is the length of the red segment, then by similar triangles you have $$ \frac r{\sqrt{(2x)^2+1}}\stackrel{(a)}=\frac x{2x}\stackrel{(b)}=\frac{x-2}1 $$ Use equation (b) to solve for $x$. Plug this into equation (a) to obtain $r$.

Answer 2

HINTS: Let the side of the square be $2x$. Calculate the slope of the blue line $m_b$, and for the red line $m_r$, in terms of $x$. Since the lines are perpendicular, $$m_b\cdot m_r=-1$$ This will allow you to find $x$. Draw the perpendicular from $O$ to the lower horizontal line, then use Pythagoras.