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

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.

b. Name another point in the diagram that would also be equidistant from points F and C. Explain how you know.

GPT-4o mini · Answer

### a. Proof that the 8-ball at point I is equidistant from holes F and C **Paragraph Proof:** To prove that the 8-ball at point I is equidistant from the holes at points F and C, we utilize the definition of a perpendicular bisector. The line segment $ GH $ is the perpendicular bisector of segment $ FC $, meaning it meets segment $ FC $ at its midpoint, J, and forms right angles with $ FC $. 1. **Definition of Perpendicular Bisector:** The definition of a perpendicular bisector states that any point on the bisector is equidistant from the endpoints of the segment it bisects. 2. Since point I lies on line segment $ GH $ (the perpendicular bisector of segment $ FC $), it meets the criteria of the definition. 3. Therefore, we conclude that the distance from point I to point F (denoted as $ d(F, I) $) is equal to the distance from point I to point C (denoted as $ d(C, I) $): $ d(F, I) = d(C, I) $. 4. This confirms that the 8-ball at point I is equidistant from the holes F and C. ### b. Name another point in the diagram that would also be equidistant from points F and C Another point that would be equidistant from points F and C is point J, the intersection point of the segment $ GH $ (the perpendicular bisector) and $ FC $. **Explanation:** Since J is defined as the midpoint of segment $ FC $, it is inherently equidistant from both endpoints F and C by the properties of a midpoint. This means $ d(F, J) = d(C, J) $, establishing that not only I, but also J is equidistant from points F and C. Each point on the perpendicular bisector $ GH $, including J, guarantees equality of distances to the endpoints of the segment $ FC $. Thus, whenever a point is on the perpendicular bisector of a segment, it is equidistant from both endpoints of that segment.