To find the area of the circle, we need to determine its radius first.

Given that chord AB is perpendicular to diameter CD, we can use the property of perpendicular chords in a circle,

which states that the product of the segments of one chord is equal to the product of the segments of the other chord.

In this case, $AD \times DB = CD \times CB$

Let's substitute the given values:

$AD \times DB = CD \times BC$

$12 cm \times DB = CD \times 8 cm$

Since AD = DB (as they are radii of the same circle), we can rewrite the equation as:

$12 cm \times AD = CD \times 8 cm$

Dividing both sides by 8 cm:

$12 cm \times AD / 8 cm = CD$

Simplifying:

$3/2 \times AD = CD$

Since CD is the diameter of the circle, we have:

$CD = 2 \times radius$

Substituting this into the equation

$3/2 \times AD = 2 \times radius$

Simplifying further:

$3 \times AD = 4 \times radius$

Now, let's substitute the given value AD = $true$

$3 \times 12 cm = 4\times radius$

$36 cm = 4\times radius$

Dividing both sides by 4:

$true$

Now that we have the radius of the circle, we can calculate its area using the formula

$A=\pi\times r^{2}:$

$A=\pi\times(9cm)^2$

$A=\pi\times81cm^{2}$

Therefore, the area of the circle is $81\pi cm^{2}$

Therefore, the correct answer is A.

Area and Volume