Since segment AB is tangent to the circle this means that angle ABC is a right angle, which in turns means that triangle ABC is a right triangle and then we can apply the pythagorean theorem:
[tex]c^2=a^2+b^2[/tex]
where c is the hypotenuse, and a and b are the legs. In this case the hypotenuse has a length of r+8 and the legs have length r and 12. Plugging this in the theorem and solving for r we have:
[tex]\begin{gathered} (r+8)\placeholder{⬚}^2=r^2+12^2 \\ r^2+16r+64=r^2+144 \\ r^2+16r-r^2=144-64 \\ 16r=80 \\ r=\frac{80}{16} \\ r=5 \end{gathered}[/tex]
Therefore, the value of r is 5
