a circle is inscribed in a rhombus with diagonals 12 and 16, what is the ratio of area of circle to that of rhombus

my doubt here is why when a perpendicular is dropped to the tangent of circle which is the side of rhombus, isnt getting bisected? This is the general rule we learn right

Generally that rule is applied plus the rule of a perpendicular bisecting any particular chord, but here we dont really need the bisection of the tangent (which is the rhombus's side), we just need to compare the areas of the triangle in 2 different ways to get the radius of the circle