Use the image to answer the question.

Question

A bird’s eye view of a pool table is shown.

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.



(4 points)

Answer 1

a. Proof that the 8-ball at point I is equidistant from points F and C:

Paragraph Proof: The line segment GH is the perpendicular bisector of segment FC. By the definition of a perpendicular bisector, it divides segment FC into two equal halves and is perpendicular to it at the midpoint J. Since point I lies on the perpendicular bisector GH, it means that the distances from I to points F and C must be equal. Therefore, we can conclude that the 8-ball located at I is equidistant from both holes F and C.

Two-Column Proof:

| Statement | Reason | |-----------|--------| | 1. GH is the perpendicular bisector of FC. | Given | | 2. J is the midpoint of FC. | Definition of a perpendicular bisector | | 3. Point I is located on GH. | Given | | 4. By definition of a perpendicular bisector, any point on GH is equidistant from F and C. | Definition of a perpendicular bisector | | 5. Thus, distance IF = distance IC. | From the definition in Statement 4 |

Consequently, Mark can confidently hit the ball towards either hole, as it is equally distant from both.

b. Another point in the diagram that would be equidistant from points F and C is point J, the midpoint of segment FC.

Explanation: Since J is the midpoint of line segment FC, by definition, it is located exactly halfway between points F and C. Thus, the distances FJ and CJ are equal, making point J equidistant from points F and C. This characteristic of midpoints is a fundamental property in geometry, confirming that segment lengths from F to J and from C to J are the same.