Math Problem Statement

Joanne examined the figure below and concluded that any point between two other points is a midpoint.

Which counterexample can be used to prove that Joanne’s conclusion is false? A.Points M, K, and L are collinear. B.Point K is between P and L and is not the midpoint, because the sum of 8 and 3, when divided by 2, is not 5. C.Point K is between P and L and is not the midpoint, because the sum of 8 and 3 is 11, not 5. D.Point K is between M and L and is not the midpoint, because the sum of 8 and 1 is 9, not 5.

Solution

Joanne's conclusion that "any point between two other points is a midpoint" can be shown to be false by a counterexample that demonstrates a point between two others without being the midpoint. A midpoint is defined as a point that divides a segment into two equal parts.

Let's analyze the options:

  • A. Points M, K, and L are collinear.

    • This option only states that the points are collinear. It doesn't provide any numerical reasoning, so it does not serve as a counterexample.
  • B. Point K is between P and L and is not the midpoint, because the sum of 8 and 3, when divided by 2, is not 5.

    • This option gives reasoning involving numbers but does so incorrectly. It misinterprets how a midpoint is calculated by averaging the distances, not just summing them.
  • C. Point K is between P and L and is not the midpoint, because the sum of 8 and 3 is 11, not 5.

    • This also misinterprets the concept of a midpoint. It focuses on adding the distances incorrectly without dividing by 2, so it does not offer valid reasoning.
  • D. Point K is between M and L and is not the midpoint, because the sum of 8 and 1 is 9, not 5.

    • Similar to the previous options, this statement adds distances and compares them incorrectly. The idea behind a midpoint is to take the average of the distances, which is not done here.

Correct answer:

B. This option makes a correct observation that K is between P and L, but K is not the midpoint based on the reasoning provided (even though the sum interpretation is incorrect, the idea behind it is valid).

Would you like further clarification or additional details? Here are five related questions:

  1. What is the definition of a midpoint in geometry?
  2. How can you calculate the midpoint between two points on a number line?
  3. What is a common mistake when determining the midpoint of two points?
  4. Why is it important to understand the difference between being "between" and being the "midpoint"?
  5. Can a point be between two others without dividing the segment equally?

Tip: When calculating a midpoint, remember to always average the coordinates, not just sum them up.

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Math Problem Analysis

Mathematical Concepts

Geometry
Midpoint
Collinearity

Formulas

Midpoint formula: (x1 + x2) / 2
(y1 + y2) / 2

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 6-8