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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¯¯¯¯¯¯¯¯ . 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)What other point on the diagram would also be equidistanct from points F and C? Explain how you know. (2 points)

1 answer

To address the questions step-by-step:

Part a: Which pocket is closer?

Since point I is on the perpendicular bisector GH¯¯¯¯¯¯¯¯ of segment FC, by the properties of a perpendicular bisector, any point on this line (such as point I) is equidistant from the endpoints F and C. Therefore:

Answer: They are the same distance from point I.

Part b: How do you know?

The theorem that supports this conclusion is the Perpendicular Bisector Theorem. This theorem states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.

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

Another point that would also be equidistant from points F and C is any point along the line segment FC itself. Specifically, the midpoint of segment FC would be one such point.

Explanation: The midpoint is the point that divides the segment into two equal parts, and thus it is equidistant from both endpoints by definition. Additionally, any point on the perpendicular bisector (line GH) is equidistant from both F and C due to the Perpendicular Bisector Theorem, as mentioned earlier.