Find the perimeter of the right triangle. The sides of the triangle are tangent to the circle.
Lines that are tangent to a circle and meet at a point are congruent, as shown in the graph:
According to the Pythagorean theorem,
\(14^2+(4+x)^2=(x+10)^2\\ 196+16+8x+x^2=x^2+20x+100\\ 112=12x\\ x=\frac{112}{12}=\boxed{\frac{28}{3}}\)
