Carlos is playing pool. He has 1 ball left to make in a pocket and then he will win. He knows that a pool table is a rectangle and there are 6 pockets that you can hit the balls into (indicated by points A, B, C, F, E, and D on the diagram below). In his head, he drew a line segment between points F and C and then pictured its perpendicular bisector, GH¯¯¯¯¯¯¯¯

Question

. The ball he needs to hit is at point I, on the perpendicular bisector. He is trying to decide which pocket is closer, F or C.

a. Which pocket is closer?

(1 point)
Responses

Pocket F is closer to Point I than Pocket C
Pocket F is closer to Point I than Pocket C

They are the same distance from point I
They are the same distance from point I

Pocket C is closer to point I than Pocket F
Pocket C is closer to point I than Pocket F
Question 2
b. How do you know? (Either give the name of the theorem or write out what it says) (2 points)
0 / 10000 Word Limit
Question 3
What other point on the diagram would also be equidistanct from points F and C? Explain how you know. (2 points)
0 / 10000 Word Limit
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Answer 1

To solve this problem, let's analyze the situation step by step.

Question 1:

a. Which pocket is closer?

Since point I lies on the perpendicular bisector (GH¯¯¯¯¯¯¯¯) of the segment connecting pockets F and C, point I is equidistant from both pockets F and C.

Answer: They are the same distance from point I.

Question 2:

b. How do you know? (Either give the name of the theorem or write out what it says)

The theorem that applies here is the Perpendicular Bisector Theorem, which states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment. Therefore, since point I is on the perpendicular bisector GH¯¯¯¯¯¯¯¯ of segment FC, point I is equidistant from both points F and C.

Answer: Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints.

Question 3:

What other point on the diagram would also be equidistant from points F and C? Explain how you know.

Any point on the line segment FC would be at varying distances from points F and C, but to find a point that is equidistant from F and C (just like point I), we can identify other points on the line that has a perpendicular bisector through I.

However, the primary point that can be guaranteed to be equidistant from F and C, besides I, is the midpoint of segment FC. This is because the midpoint of a line segment is defined as the point that divides the segment into two equal parts. It lies directly on the line joining F and C and is equidistant from both points.

Answer: The midpoint of segment FC is also equidistant from points F and C, because it is the point that divides the segment into two equal lengths.