We are given the common tangent of two circles. To determine the value of AB we need to form a right triangle and apply the Pythagorean theorem. The triangle will be formed by segment OP as the hypotenuse and the segment from O to the point where the perpendicular line to OA that passes through P as one of its sides. The length of this side is:
[tex]\begin{gathered} b=OA-PB \\ b=8-2=6 \end{gathered}[/tex]
The value of segment OP is:
[tex]OP=8+2=10[/tex]
Applying the theorem:
[tex]h=\sqrt[]{OP^2-b^2}[/tex]
Replacing the values:
[tex]\begin{gathered} h=\sqrt[]{10^2-8^2} \\ h=\sqrt[]{100-64} \\ h=\sqrt[]{36} \\ h=6 \end{gathered}[/tex]
Since this segment has the same length as AB we have:
[tex]AB=6[/tex]
