CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The midpoint of the segment joining (0, 0) and (4, 4) is ________.

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Recall that the midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[\left( \frac{\,x_1 + x_2}{2}, \frac{\,y_1 + y_2}{2} \right)\]

Identify the coordinates of the two given points: \((0, 0)\) and \((4, 4)\). Here, \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 4\), and \(y_2 = 4\).

Substitute these values into the midpoint formula: \[\left( \frac{0 + 4}{2}, \frac{0 + 4}{2} \right)\]

Simplify the expressions inside the parentheses by performing the addition and division: \[\left( \frac{4}{2}, \frac{4}{2} \right)\]

Express the simplified midpoint coordinates as a point: \[\left( 2, 2 \right)\]

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주요 개념

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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 of the segment.

Coordinate Geometry

Coordinate geometry involves representing geometric figures using coordinates on the Cartesian plane. Understanding how points, lines, and segments are expressed in (x, y) form is essential for applying formulas like the midpoint.

Application of Arithmetic Operations

Calculating the midpoint requires basic arithmetic operations such as addition and division. Being comfortable with these operations ensures accurate computation of the average coordinates.