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 overline GH has been drawn as the perpendicular bisector of overline PC. overline GH intersects overline PC at point J. Mark only has one ball left to he 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 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

Answer 1

To prove that the 8-ball located at point I is equidistant from the hole at point F and the hole at point C using a paragraph proof:

Proof:

The line segment GH is defined as the perpendicular bisector of line segment PC. By definition, a perpendicular bisector divides a line segment into two equal parts and is perpendicular to the segment at its midpoint. Since point J is the point of intersection of line segment GH and line segment PC, it represents the midpoint of segment PC. Therefore, we have \(PJ = JC\).

Since point I lies on line segment GH, which is the perpendicular bisector of line segment PC, it is also true that point I is equidistant from points P and C. Consequently, because line segment GH bisects segment PC at a right angle, it follows that the distances from point I to points F and C are equal: \(IF = IC\). Hence, we can conclude that the 8-ball located at point I is equidistant from both hole F and hole C.

b.

Another point in the diagram that would also be equidistant from points F and C is point J, where line segments GH and PC intersect.

We know this because point J is the intersection of the perpendicular bisector (line segment GH) and segment PC. By the properties of perpendicular bisectors, any point on the bisector (including point J) is equidistant from both endpoints of the segment it bisects (in this case, points F and C). Therefore, point J is also equidistant from points F and C, just as point I is.