Number 19, consider a convex hexagon with sides tangent to an inscribed circle. Five consecutive sides have the lengths 4, 5, 6, 7, and 8, respectively, as I have drawn in here, and we see that one side has a missing dimension. We have to compute for this question the length of the sixth side. So in order to get started, we will have to label some of the sides. And before we do that, we note that drawing in a line segment between any two consecutive points of tangency, like so, creates a triangle that is, in fact, isosceles, with these two angles being the same, therefore, two sides of equal length. So when we label these, and I'll begin here with A, I can use the same label again, and then going counterclockwise here, I would have a B, C, and C, D twice, E twice, and finally F. So we see that the missing side has length equal to exactly E plus F, and that's what we will try to compute. So we note that F also appears in the side labeled with an 8, E appears with the side labeled with a 4, so we write down some equations, 4 plus 8 is equal to 12, and that is going to be E plus D plus A plus F, so we see that 12 is equal to E plus F plus A plus D, E plus F is what we care about, we have to somehow subtract A plus D from both sides, so let's see what those two lengths add up to, A appears inside 7, and D appears inside 5, so we can add those together to obtain A plus D, and so we will have here 7 plus 5, that's also equal to 12, that's equal to the sum of the letters A plus B plus C plus D, so 12 is equal to A plus D plus B plus C, and we note that B plus C is exactly 6, so 6, by subtracting 6 from both sides is A plus D, so finally with this information, with A plus D being equal to 6, we have that 6 is equal to E plus F, so we have found our answer, and the answer is choice D.

convex hexagon

inscribed circle

missing side length

tangent points

geometry problem