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.
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 properties of perpendicular bisectors.
First, let's consider the perpendicular bisector GH¯¯¯¯¯¯¯¯. Since GH¯¯¯¯¯¯¯¯ is the perpendicular bisector of FC¯¯¯¯¯¯¯¯, this means that GH¯¯¯¯¯¯¯¯ intersects FC¯¯¯¯¯¯¯¯ at a right angle and divides it into two equal segments.
Let's call the intersection point of GH¯¯¯¯¯¯¯¯ and FC¯¯¯¯¯¯¯¯ as J. Since GH¯¯¯¯¯¯¯¯ is the perpendicular bisector, we can say that IJ¯¯¯¯¯¯¯¯ is equal to JF¯¯¯¯¯¯¯¯ and also equal to JC¯¯¯¯¯¯¯¯. This is because J is equidistant from F and C, as it lies on the perpendicular bisector.
Now, let's prove that IF¯¯¯¯¯¯¯¯ is equal to IC¯¯¯¯¯¯¯¯.
We know that JF¯¯¯¯¯¯¯¯ is equal to JC¯¯¯¯¯¯¯¯. Also, as J lies on the perpendicular bisector GH¯¯¯¯¯¯¯¯, we can say that IJ¯¯¯¯¯¯¯¯ is equal to JF¯¯¯¯¯¯¯¯ and JC¯¯¯¯¯¯¯¯.
Therefore, by transitive property, IJ¯¯¯¯¯¯¯¯ is equal to IF¯¯¯¯¯¯¯¯ and IC¯¯¯¯¯¯¯¯.
Hence, the 8-ball at point I is equidistant from the hole at point F and the hole at point C.
b. Another point in the diagram that would also be equidistant from points F and C is the midpoint of segment FC¯¯¯¯¯¯¯¯. Let's call this point M.
Since M is the midpoint of FC¯¯¯¯¯¯¯¯, we know that MF¯¯¯¯¯¯¯¯ is equal to MC¯¯¯¯¯¯¯¯. Additionally, as F and C are the corner holes on the pool table, they are equidistant from the midpoint of their respective long sides (FC¯¯¯¯¯¯¯¯). Therefore, MF¯¯¯¯¯¯¯¯ is also equal to IF¯¯¯¯¯¯¯¯ and IC¯¯¯¯¯¯¯¯.
Hence, point M is equidistant from points F and C. This can be concluded based on the properties of midpoints and the fact that F and C are located symmetrically on the long sides of the pool table.