Question

I need help on in the answer to this question

Answer 1

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:

`LM\cdot LN=(KL)^2`

Replace KL=18, LM=12 and LN=12+(2x-9) in order to get an equation for x and find its value:

`\Rightarrow12(12+2x-9)=18^2`

Solve for x:

`\begin{gathered} \Rightarrow12(2x+3)=324 \\ \Rightarrow2x+3=(324)/(12) \\ \Rightarrow2x+3=27 \\ \Rightarrow2x=27-3 \\ \Rightarrow2x=24 \\ \Rightarrow x=(24)/(2) \\ \therefore x=12 \end{gathered}`

Replace x=12 in the expression for MN to find the length MN:

`\begin{gathered} MN=2x-9 \\ =2(12)-9 \\ =24-9 \\ =15 \end{gathered}`

Therefore, the measure of MN is 15.

The correct choice is option C) 15.