Divide the dartboard into six sectors by drawing a set of six radii spaced 60 degrees apart.  Each of the six sectors fits inside a circle whose diameter equals the radius of the dartboard.  If two points are in the same sector then they must be at most one dartboard radius apart.

Part A is solved by applying the pigeonhole principle.  With 7 points and 6 sectors at least two points must be in the same sector and therefore within one dartboard radius of each other.

For part B, draw the radii of the dartboard containing each of the six points.  At least one pair of consecutive radii must have a subtended angle of no more than 60 degrees.  Then rotate the placement of the six sectors until this pair of radii are in the same sector.  Then the corresponding points must be within one dartboard radius of each other.