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.
a. Two points are equidistant from a line if and only if they lie on the perpendicular bisector of that line. In this case, point J lies on the perpendicular bisector GH¯¯¯¯¯¯¯¯ of line segment FC¯¯¯¯¯¯¯¯. Therefore, J is equidistant from F and C.
Since point I lies on the perpendicular bisector GH¯¯¯¯¯¯¯¯, we know that it is equidistant from F and C. This is because any point on the perpendicular bisector of a line segment is equidistant from the endpoints of that line segment. Therefore, 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. As mentioned in part a, point J lies on the perpendicular bisector GH¯¯¯¯¯¯¯¯ of line segment FC¯¯¯¯¯¯¯¯. Therefore, J is equidistant from F and C.