Question

Use the image to answer the 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 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)BoldItalicUnderlineIncrease indentDecrease indentFormula keypadUndo

Answer 1

To address your request, I will provide a proof regarding the position of the 8-ball and identify another point that is equidistant from points F and C.

Part A: Proof that point I is equidistant from points F and C

Let's use a paragraph proof:

Paragraph Proof:

1. Definition of Perpendicular Bisector: A perpendicular bisector of a line segment is a line that is perpendicular to the segment at its midpoint and divides the segment into two equal parts.
2. Given: Line segment \(GH\) is the perpendicular bisector of line segment \(FC\). By definition, this means that \(J\), the point where \(GH\) intersects \(FC\), is the midpoint of segment \(FC\).
3. Location of Point I: Point \(I\) is located on the perpendicular bisector \(GH\).
4. Properties of Perpendicular Bisector: Since point \(I\) lies on the perpendicular bisector of segment \(FC\), it is equidistant from both endpoints \(F\) and \(C\).
5. Conclusion: Therefore, the distance from point \(I\) to point \(F\) is equal to the distance from point \(I\) to point \(C\) (\(IF = IC\)). As a result, the 8-ball at point \(I\) is equidistant from both holes at points \(F\) and \(C\).

Part B: Identify another point equidistant from F and C

Another point that is also equidistant from points \(F\) and \(C\) is point \(J\).

Explanation:

1. Position of Point J: Point \(J\) is defined as the intersection of the line segment \(GH\) (the perpendicular bisector) and the segment \(FC\).
2. Definition of Midpoint: By the properties of a perpendicular bisector, \(J\) is the midpoint of segment \(FC\).
3. Equidistant Property: Since \(J\) is the midpoint, it is equidistant from both points \(F\) and \(C\) (\(JF = JC\)), satisfying the condition of having equal distances to both points.

Thus, point \(J\) is another point that is equidistant from points \(F\) and \(C\).