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;
[tex]AC^2=AB^2+BC^2[/tex]
Substituting the values, we have;
[tex][(r+1)+r]^2=(2r-1)^2+r^2[/tex]
Simplifying, we have;
[tex](2r+1)^2=(2r-1)^2+r^2[/tex]
Expanding the terms, we get;
[tex]4r^2+4r+1=4r^2-4r+1+r^2[/tex]
[tex]4r^2+4r+1=5r^2-4r+1[/tex]
Simplifying the values, we have;
[tex]4r=-4r+r^2[/tex]
Adding both sides of the equation by 4r, we get;
[tex]8r=r^2[/tex]
[tex]0=r^2-8r[/tex]
[tex]0=r(r-8)[/tex]
Thus, [tex]r=0 \ or \ r=8[/tex]
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,
[tex]C=2 \pi r[/tex]
Substituting r = 8, we get;
[tex]C=2 \pi (8)[/tex]
[tex]C=16 \pi[/tex]
Thus, the circumference of the circle is 16π
Hence, Option A is the correct answer.
