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 FC¯¯¯¯¯¯¯¯ 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 I 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

a. Proof Using Perpendicular Bisector

To prove that the 8-ball at point I is equidistant from holes F and C, we can use the properties of perpendicular bisectors.

Paragraph Proof: Let F and C be the two hole locations on the pool table. Since GH is the perpendicular bisector of line segment FC, it has specific properties:

1. By definition, a perpendicular bisector divides a segment into two equal halves and is perpendicular to that segment.

   • Therefore, \( \overline{JF} = \overline{JC} \) (segments are equal because GH is the perpendicular bisector).

2. The point I lies on the line GH, which is the perpendicular bisector of segment FC.

   • By the property of the perpendicular bisector, any point on GH is equidistant from points F and C.

3. Since I lies on the perpendicular bisector GH, we conclude:

   • \( \overline{IF} = \overline{IC} \), meaning point I is equidistant from points F and C.

Thus, we have proven that the 8-ball located at point I is equidistant from holes F and C.

b. Another Point Equidistant from F and C

Another point that would be equidistant from points F and C is point J, which is the intersection point of the perpendicular bisector GH and line segment FC.

Explanation:

1. Since J lies on the perpendicular bisector GH, it has the same property that any point on the perpendicular bisector is equidistant from points F and C.
2. Therefore, \( \overline{JF} = \overline{JC} \), confirming that point J is equidistant from both holes.

In conclusion, points I (where the 8-ball is) and J (the intersection of the perpendicular bisector and segment FC) are both equidistant from holes F and C.