Given:
PO is the perpendicular bisector of triangle MON.
[tex]\begin{gathered} MP=2x+5 \\ \angle OPN=5x+10 \end{gathered}[/tex]1. To find the value of x:
Since,
[tex]OP\perp MN[/tex]Therefore, the angle measure of OPN is 90 degrees.
So that,
[tex]\begin{gathered} \angle OPN=90^{\circ} \\ 5x+10=90 \\ 5x=80 \\ x=16 \end{gathered}[/tex]Hence the value of x is 16.
2. To find MN:
Since PO is the perpendicular bisector to the side MN.
So, MP=PN
Therefore,
[tex]\begin{gathered} MN=MP+PN \\ =MP+MP \\ =2MP \\ =2(2x+5) \\ =4x+10 \\ =4(16)+10\text{ \lbrack{}Substituting x = 16\rbrack} \\ =64+10 \\ MN=74 \end{gathered}[/tex]Hence, the length of MN is 74.