Question

Angle measures and segment lengths. Someone please help me and explain

Answer 1

The intersecting secants theorem says

$6(6+12)=5(5+x)$

That is, for either secant line, you take the length of the part of the secant line outside the circle (6 and 5 in this case) and multiply it by the "total" length of the secant line between the point where the two secant lines intersect and the furthest point where the secant touches the circle (6+12=18 and 5+x in this case). The theorem says these products are equal.

Then

$6\cdot18=25+5x$

$5x=83$

$x=\dfrac{83}5=\boxed{16.6}$