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.

Question

Line segment overline GH has been drawn as the perpendicular bisector of overline FC. Overline GH intersects overline 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 overline 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 point C. You can choose if you want to write a paragraph in a 7th graders response 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.

Write in 3 sentences 7th graders response

Answer 1

a. To prove that the 8-ball at point I is equidistant from the hole at point F and the hole at point C, we can use the property of a perpendicular bisector. Since line segment GH is the perpendicular bisector of segment FC, it means that any point on line GH, including point I, is the same distance from both endpoints F and C. Therefore, the distance from point I to point F is equal to the distance from point I to point C, proving that the 8-ball is equidistant from both holes.

b. Another point that would be equidistant from points F and C is point J, where line segment GH intersects line segment FC. We know this because point J lies on the perpendicular bisector of FC, which means it is also equally distant from both F and C, just like point I. This property of perpendicular bisectors tells us that any point on it will always have the same distance to the endpoints of the segment it bisects.