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{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 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.

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{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 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.

Accepted Answer

# Explanation:
## Step1: State the perpendicular bisector theorem
The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
## Step2: Apply the theorem to point $I$
Since $GH$ is the perpendicular bisector of $FC$ and point $I$ lies on $GH$, by the perpendicular bisector theorem, $IF = IC$. So the 8 - ball at point $I$ is equidistant from the hole at point $F$ and the hole at point $C$.

## Step3: Find another equidistant point
Another point that is equidistant from $F$ and $C$ is point $J$.
## Step4: Justify for point $J$
Since $J$ is the intersection of $GH$ (the perpendicular bisector of $FC$) and $FC$, by the definition of a perpendicular bisector (a point on the perpendicular bisector), $JF=JC$

# Answer:
a. By the perpendicular bisector theorem, since $GH$ is the perpendicular bisector of $FC$ and $I$ is on $GH$, $IF = IC$.
b. Point $J$ is equidistant from $F$ and $C$ because $J$ is on the perpendicular bisector $GH$ of $FC$ (by the perpendicular bisector theorem, a point on the perpendicular bisector of a segment is equidistant from the segment's endpoints).

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QUESTION IMAGE

Question

Explanation:

Step1: State the perpendicular bisector theorem

Step2: Apply the theorem to point \(I\)

Step3: Find another equidistant point

Step4: Justify for point \(J\)

Answer: