Question

Not sure I understand this...

Answer 1

I know that the closest point is the midpoint of chord PQ, but I don't know why.

Answer 2

yep; as the earlier replier said.

"Of all the points joining a line to another point, the perpendicular is the shortest."

Hence the _|_ line joins mid point of PQ to center as the closest point on the chord to the center. = ans.

Answer 3

the triangle formed by OPQ (O at the circle's center), is isosceles
(two legs formed by radii)

We can prove that the altitude of the triangle from point O is the perpendicular bisector of side PQ (chord PQ)

Answer 4

is what it looks like to me

Answer 5

Yes...that's what it looks like but I son't know how to say why it is shorter.

Answer 6

well, I'd probably say, because is the farthest point FROM THE ARC in the chord PQ, the farther from the ARC, the closer to the CENTER. PQ's perpendicularity makes it 90degrees on either side of the bisector at that point, and thus the shortest, and thus the closest to any reference point on the bisector.