Determine whether each statement is true or false. If false, explain why. The midpoint of the segment joining (0, 0) and (4, 4) is 2.

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Recall the formula for the midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\): \[\text{Midpoint} = \left( \frac{\,x_1 + x_2}{2}, \frac{\,y_1 + y_2}{2} \right)\]

Identify the given points: \((0, 0)\) and \((4, 4)\). Substitute these into the midpoint formula: \[\left( \frac{0 + 4}{2}, \frac{0 + 4}{2} \right)\]

Simplify each coordinate: \[\left( \frac{4}{2}, \frac{4}{2} \right) = (2, 2)\]

Analyze the statement: it claims the midpoint is 2, which is a single number, not a point with two coordinates. Since the midpoint is \((2, 2)\), the statement is false.

Explain why: The midpoint of a segment in the coordinate plane is always a point with two coordinates, not a single number. Therefore, saying the midpoint is 2 is incorrect.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Midpoint Formula

The midpoint of a line segment between two points (x₁, y₁) and (x₂, y₂) is found by averaging their coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). This gives the exact center point on the segment.

Coordinate Points in the Plane

Points in the coordinate plane are represented as ordered pairs (x, y). Understanding how to interpret and manipulate these pairs is essential for calculating distances, midpoints, and other geometric properties.

Interpreting Statements About Points

Statements about points must be precise; for example, a midpoint is a point with two coordinates, not a single number. Recognizing the difference between a point and a scalar value is crucial for evaluating the truth of such statements.