Mark is playing pool. On a pool table there are 6 holes that you have to hit the balls into. Four of the holes are located at the four corners of the table, and the other two holes are located at the midpoints of the long sides of the table. These two holes are indicated on the image at points F and C, and a line segment has been drawn connecting these two points. Line segment GH has been drawn as the perpendicular bisector of FC. GH intersects F'C' at point J. Mark only has one ball left to hit in, the 8-ball. The 8-ball is located at point I, which is on perpendicular bisector GH. Mark is trying to decide if the 8-ball is closer to the hole at point F or the hole at point C.

Question

a. Use your knowledge about perpendicular bisectors to prove to Mark that the 8-ball at point / is equidistant from the hole at point F and the hole at point C. You can choose if you want to write a paragraph proof or a two-column proof, but be sure to include a reason for each of your statements.
b. Name another point in the diagram that would also be equidistant from points F and C.
Explain how you know.

Answer 1

Part a: Proof that the 8-ball is equidistant from holes F and C

Paragraph Proof:

To prove that the point I (where the 8-ball is located) is equidistant from points F and C (the holes), we can use the property of perpendicular bisectors. Since GH is the perpendicular bisector of segment FC, by definition, any point that lies on the perpendicular bisector is equidistant from the endpoints of the segment it bisects. Therefore, since point I lies on perpendicular bisector GH, we conclude that the distance from point I to point F is equal to the distance from point I to point C. Thus, we can write:

1. GH is the perpendicular bisector of segment FC.
2. By definition of a perpendicular bisector, any point on it is equidistant from the endpoints of the segment it bisects.
3. Since point I is on the perpendicular bisector GH, it is equidistant from points F and C.
4. Therefore, the distance IF (distance from point I to point F) is equal to the distance IC (distance from point I to point C): IF = IC.

Hence, Mark can conclude that the 8-ball at point I is equidistant from holes F and C.

Part b: Another point equidistant from points F and C

Another point that would also be equidistant from points F and C is the midpoint of segment FC.

Explanation:

The midpoint of a segment is defined as the point that is exactly halfway between the two endpoints of that segment. In this case, if we let the midpoint of segment FC be denoted as point M, then by the definition of a midpoint, point M is at equal distance from both endpoints F and C. Therefore, the distances FM and CM are equal, symbolically written as FM = CM.

Since the midpoint M lies directly on the segment FC, it can also be observed that if M is on GH (the perpendicular bisector of FC), which it must be, then it further reinforces the concept that any point on the perpendicular bisector (including M) remains equidistant from F and C.