Question

Given AB is tangent to circle C at point B, what is the circumference of circle C?

Answer 1

Given:

Given that AB is a tangent to circle C.

The length of AB is (2r -1)

The length of AC is (r + 1) + r

The length of BC is r.

We need to determine the circumference of the circle C.

Value of r:

The value of r can be determined using the Pythagorean theorem.

Thus, we have;

$AC^2=AB^2+BC^2$

Substituting the values, we have;

$[(r+1)+r]^2=(2r-1)^2+r^2$

Simplifying, we have;

$(2r+1)^2=(2r-1)^2+r^2$

Expanding the terms, we get;

$4r^2+4r+1=4r^2-4r+1+r^2$

$4r^2+4r+1=5r^2-4r+1$

Simplifying the values, we have;

$4r=-4r+r^2$

Adding both sides of the equation by 4r, we get;

$8r=r^2$

 $0=r^2-8r$

 $0=r(r-8)$

Thus, $r=0 \ or \ r=8$

Since, the radius of the circle cannot be 0.

Hence, the radius of the circle is 8.

Circumference of the circle:

The circumference of the circle can be determined using the formula,

$C=2 \pi r$

Substituting r = 8, we get;

$C=2 \pi (8)$

$C=16 \pi$

Thus, the circumference of the circle is 16π

Hence, Option A is the correct answer.