Answer:
x = 13 ; MN = 71
Step-by-step explanation:
NP is perpendicular bisector of OM.
=> MP = OP
Also here we get 2 right angled triangles named MPN & OPN.
In triangle MPN ,
[tex] {MN}^{2} = {MP}^{2} + {PN}^{2} [/tex]
[tex] = > {MN}^{2} = {OP}^{2} + {PN}^{2} [/tex]
because MP = OP.
In triangle OPN ,
[tex] {ON}^{2} = {OP}^{2} + {PN}^{2} [/tex]
But
[tex] {OP}^{2} + {PN}^{2} = {MN}^{2} [/tex]
Hence ,
[tex] {OP}^{2} = {MN}^{2} [/tex]
[tex] = > OP = MN[/tex]
Value of MN = 4x + 19
Value of ON = 6x - 7
So,
[tex]4x + 19 = 6x - 7[/tex]
[tex] = > 6x - 4x = 19 + 7[/tex]
[tex] = > 2x = 26[/tex]
[tex] = > x = \frac{26}{2} = 13[/tex]
So MN = 4×13 + 19 = 71
