Let Angle MLN = x ;
Angle KLQ = 90 - x (supplementary Angles )
in Triangle LMN , Angle LMN = 90- x ( Angle sum of Triangle )
in triangle QKL, Angle QKL = x ( Angle sum of trangle )
....
Angle MLN = Angle QKL ( x )
Angle LMN = Angle KLQ ( 90 - x )
Angle MNL = Angle KQL ( right angles given )
Hence.. triangles are similar as they are equiangular
To get y apply the fact that sides in similar triangles are in the same ratios
y / MN = KL / QL
We know KL = 6 + x and QL = root 24x = 2root(6x) and MN = 12
y / 12 = 6+ x / 2root (6x)
y = 12(6+x) / 2root(6x) ( 12/2 comes down to 6/1 ; root(6x) = root 6 x root x )
y = 6/ (6+ x ) / root(6) x root(x)
y = root 6 (6+x) / root(x) (as 6 / root6) is root6 )
