We are given
tangent line as
[tex] LK=36 [/tex]
we have
[tex] LM=27 [/tex]
[tex] LN=LM+MN [/tex]
[tex] LN=27-3+3x [/tex]
[tex] LN=3x+24 [/tex]
now, we can use formula
[tex] LK^2=LM*LN [/tex]
now, we can plug values
[tex] 36^2=27*(3x+24) [/tex]
now, we can solve for x
step-1: Divide both sides by 27
[tex] \frac{36^2}{27} =3x+24 [/tex]
[tex] 3x+24=48 [/tex]
step-2: Subtract both sides by 24
[tex] 3x+24-24=48-24 [/tex]
[tex] 3x=24 [/tex]
step-3: Divide both sides by 3
[tex] \frac{3x}{3} =\frac{24}{3} [/tex]
[tex] x=8 [/tex]
Calculation of MN:
We know that
[tex] MN=-3+3x [/tex]
now, we can plug x=8
[tex] MN=-3+3*8 [/tex]
[tex] MN=21 [/tex]....................Answer
