The problem involves finding the lengths x and y in a right triangle ABC using the Pythagorean theorem. After calculations, x + y is found to be approximately 16.1. So, the correct option is B.
The problem presents a right triangle which we name ABC where AC is the hypotenuse, AD bisects BC at D making AD perpendicular to BC, and BD=9, CD=3, AD=5. The lengths of AB and AC are represented by x and y respectively, and we need to find x + y.
First, notice that triangle ABC is a right triangle with AD as the altitude from the right angle at A. Since D is the midpoint of BC, triangles ABD and ACD are similar to triangle ABC and to each other.
Given BD=9 and CD=3, and by the Pythagorean theorem in triangle ABD, we can find x (AB) using x2 = AD2 + BD2, thus x2 = 52 + 92, resulting in x2 = 25 + 81, x2 = 106, and x = sqrt(106), which is approximately 10.3cm.
Using the same reasoning for triangle ACD, we find y (AC) using the equation y2 = AD2 + CD2, thus y2 = 52 + 32, y2 = 25 + 9, y2 = 34, and y = sqrt(34), which is approximately 5.8cm. Finally, x + y = 10.3 + 5.8 = 16.1.