Two parallel chords in a circle of radius 7.5cm measure 12cm and 9cm. They are on the same side of the circle. What is the distance between

Here's one method.....but.....maybe not the fastest one !!!

Let the center of the circle lie at the origin and let both chords be parallel to the x axis.

The equation of the circle  is  x^2 + y^2   = 56.25

The chord of 12 cm in  length will have one of its endpoints at  (6,y₁)......and y₁ =

√[56.25 - 6^2] = √[56.25 -36 ]  = 4.5  { taking the positive value of the  root  }

And

The chord of 9 cm in  length will have one of its endpoints at  (4.5, y₂)......and y₂ =

√[56.25 - 4.5^2] = √[56.25 - 20.25 ]  = 6   {again, taking the positive value of the root}

So....the distance between the chords is  y₂ - y₁  = 6 - 4.5   = 1.5 cm

Here's a pic.....

Note that ED = 9, CB = 12 and  FG = 1.5