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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 foru 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 the 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 / is equidistant from the hole at point F and the hole at 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

1 answer

a. Paragraph Proof

To prove that the 8-ball at point I is equidistant from the holes at points F and C, we can use the properties of the perpendicular bisector. By definition, a perpendicular bisector of a segment divides the segment into two equal halves and is perpendicular to that segment. In this case, segment FC is bisected by line segment GH at point J, meaning that:

  1. Point J is the midpoint of segment FC.
  2. Therefore, segments FJ and JC are equal in length (FJ = JC).
  3. Since point I lies on the perpendicular bisector GH, any point on a perpendicular bisector is equidistant to the endpoints of the segment it bisects.
  4. Thus, the distance from point I to point F is equal to the distance from point I to point C (IF = IC).

Hence, we can conclude that the 8-ball is equidistant from the holes at points F and C.

b. Another Equidistant Point

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

This is because:

  1. As stated earlier, J is the midpoint of FC, which means that it is located exactly halfway between F and C.
  2. By the properties of a bisector, we know that the distances from the midpoint to each endpoint are equal (FJ = JC).
  3. Therefore, point J, like point I, is also equidistant from points F and C.

Marks can use this property of midpoints and perpendicular bisectors to understand that there are indeed multiple points, such as I and J, that are equidistant from F and C.