AB is the chord passing through the center,

So, AB is the diameter

Since angle in a semi-circle is a right angle

∠APB= 90°

By using alternate segment theorem

We have ∠APB = ∠PAT = 30°

Now, in ⧍APB

∠BAP + ∠APB + ∠BAP = 180′ (Angle sum property of triangle)

∠BAP = 180° – 90° – 30° = 60°

Now, ∠BAP = ∠APT + ∠PTA (Exterior angle property)

60° = 30° + ∠PTA

∠PTA = 60° – 30° = 30°

We know that sides opposite to equal angles are equal

AP = AT

In right triangle ABP:

=> sin ABP=AP/BA

=> sin30° = AT/BA   {since AP = AT}

=>1/2 = AT/BA

=> BA/AT = 2/1

=> BA : AT = 2 : 1