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 a 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 .

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 a 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 I is equidistant from the hole at point F and the hole at point C . Using the dropdown menus, fill out the blanks for the proof below.

1. GH¯¯¯¯¯¯¯¯ is the perpendicular bisector of FC¯¯¯¯¯¯¯¯ 1.
2. FJ¯¯¯¯¯¯¯≅CJ¯¯¯¯¯¯¯ 2. Definition of a Perpendicular Bisector
3. ∠IJF and ∠IJC are right angles 3.
4. ∠IJF≅∠IJC 4. Right Angles Congruence Theorem
5. IJ¯¯¯¯¯¯≅IJ¯¯¯¯¯¯ 5.
6. △IJF≅△IJC 6.
7. IF¯¯¯¯¯¯≅IC¯¯¯¯¯¯ 7.

Answer 1

1. GH¯¯¯¯¯¯¯¯ is the perpendicular bisector of FC¯¯¯¯¯¯¯¯ 1. Definition of a perpendicular bisector: A line that intersects a segment at its midpoint and forms right angles with the segment.

2. FJ¯¯¯¯¯¯¯≅CJ¯¯¯¯¯¯¯ 2. If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.
3. ∠IJF and ∠IJC are right angles 3. By definition, a line is perpendicular to another line if it forms a 90-degree angle with it.
4. ∠IJF≅∠IJC 4. If two angles are congruent to a third angle, then they are congruent to each other.
5. IJ¯¯¯¯¯¯≅IJ¯¯¯¯¯¯ 5. Reflexive property of congruence: Any segment is congruent to itself.
6. △IJF≅△IJC 6. If two angles are congruent, and the side between them is congruent, then the triangles are congruent.
7. IF¯¯¯¯¯¯≅IC¯¯¯¯¯¯ 7. Corresponding parts of congruent triangles are congruent. Since △IJF≅△IJC, then the corresponding sides are also congruent.