12.8 is the answer in the book, I will avoid calling it the correct answer until I can find a way to calculate it without making an assumption. The method I use to get this answer involves assuming that the bigger triangle is a right triangle (i.e. a triangle with a 90degree angle in it). This is important because Pythagoras theorem only applies to right triangles.
to get 12.8, I noticed that the 2 tangents to the circle have a common point. The tangents are the side with length 8 and the unlabelled side of the bigger triangle. (i.e. the bigger triangle has a side labelled length 10, and another x, then the third is unlabelled, I will call it y)
the side with length 8 equals the length of side y because they are tangent segments that extend from the same point.
If the bigger triangle is a right triangle, we can find x using the Pythagoras theorem, 10^2+8^2 =164
square root(164) = 12.8
I think you highlighted the exact problem with assuming that the bigger triangle is a right triangle. I say it is a right triangle and you say it looks more like an obtuse triangle and someone else might call it isosceles or equilateral. Unless someone can provide a reason that is more precise than that's what it looks like we cannot be sure which one is correct. The drawings are also not to scale so going by looks is unreliable. this is also why we cannot use a protractor/ruler to just measure the sides.