The given situation can be illustrated as follow:
The blue lines represent the mirrors.
Based on the given information you have:
AB = 9 + AC
BC = AC - 9
By using the Pythagorean theorem you can write:
[tex](AB)^2=(BC)^2+(AC)^2[/tex]Replace the given expressions for AB and BC into the previous equation and solve for AC, as follow:
[tex]\begin{gathered} (9+AC)^2=(AC-9)^2+(AC)^2 \\ 81+2\cdot9\cdot AC+(AC)^2=(AC)^2-2\cdot9(AC)+81+(AC)^2 \\ 18AC+18AC=(AC)^2 \\ 36AC=(AC)^2 \end{gathered}[/tex]Where we have expanaded (9+AC)^2 and (AC-9)^2. By dividing by AC both sides:
[tex]\begin{gathered} 36=AC \\ AC=36 \end{gathered}[/tex]Hence, the distance between mirrors A and C is 36
