To find NE when N is the incenter of triangle ABC we use the fact that the incenter is equidistant from all sides to set ND equal to NE, solve for x, and substitute x back into the expression for NE, which gives us NE = 19.
The question involves finding the length of segment NE when point N is the incenter of triangle ABC, and segment lengths ND and NF are given in terms of x. To solve this, we set the two expressions equal to each other because the incenter of a triangle is equidistant from all sides, and therefore ND is equal to NE as they are both radii to tangents from a point outside a circle.
So we have:
ND = NE
5x + 4 = 2x + 13
We then solve for x:
5x - 2x = 13 - 4
3x = 9
x = 3
Now we can find NE (ND) by substituting x back into either expression:
NE = 5x + 4
NE = 5(3) + 4
NE = 15 + 4
NE = 19