According to the Tangent-Secant theorem, the length of the tangent segment KL squared, must be equal to the product of the lengths of the secant segments LN and LM:
[tex]LM\cdot LN=(KL)^2[/tex]
Replace KL=18, LM=12 and LN=12+(2x-9) in order to get an equation for x and find its value:
[tex]\Rightarrow12(12+2x-9)=18^2[/tex]
Solve for x:
[tex]\begin{gathered} \Rightarrow12(2x+3)=324 \\ \Rightarrow2x+3=\frac{324}{12} \\ \Rightarrow2x+3=27 \\ \Rightarrow2x=27-3 \\ \Rightarrow2x=24 \\ \Rightarrow x=\frac{24}{2} \\ \therefore x=12 \end{gathered}[/tex]
Replace x=12 in the expression for MN to find the length MN:
[tex]\begin{gathered} MN=2x-9 \\ =2(12)-9 \\ =24-9 \\ =15 \end{gathered}[/tex]
Therefore, the measure of MN is 15.
The correct choice is option C) 15.
