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. 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.

Question

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. 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. Two points on a perpendicular bisector are equidistant from the endpoints of the line segment. In this case, since I is on the perpendicular bisector GH¯¯¯¯¯¯¯¯, it is equidistant from points F and C. This can be proven as follows:

Statement 1: GH¯¯¯¯¯¯¯¯ is the perpendicular bisector of FC¯¯¯¯¯¯¯¯
Reason 1: Given

Statement 2: J is the point of intersection of GH¯¯¯¯¯¯¯¯ and FC¯¯¯¯¯¯¯¯
Reason 2: Given

Statement 3: GH¯¯¯¯¯¯¯¯ bisects FC¯¯¯¯¯¯¯¯
Reason 3: Definition of a perpendicular bisector

Statement 4: JF¯¯¯¯¯¯¯¯ is congruent to JC¯¯¯¯¯¯¯¯
Reason 4: Perpendicular bisector theorem

Statement 5: JF¯¯¯¯¯¯¯¯ is congruent to JI¯¯¯¯¯¯¯¯
Reason 5: Given

Statement 6: JC¯¯¯¯¯¯¯¯ is congruent to JI¯¯¯¯¯¯¯¯
Reason 6: Transitive property of congruence

Statement 7: J is equidistant from F and C
Reason 7: Definition of congruent line segments

Statement 8: I is equidistant from F and C
Reason 8: Definition of equidistant

Therefore, we have proven that the 8-ball at point I is equidistant from the hole at point F and the hole at point C.

b. Another point in the diagram that would also be equidistant from points F and C is point J. This can be deduced from the fact that J is the point of intersection of the perpendicular bisector GH¯¯¯¯¯¯¯¯ and the line segment FC¯¯¯¯¯¯¯¯. Since the perpendicular bisector divides the line segment into two congruent segments, J is equidistant from both endpoints, F and C.