Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-1/4, -1/7) and (3/4, 6/7)

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Identify the coordinates of the two points: \(\left(-\frac{1}{4}, -\frac{1}{7}\right)\) and \(\left(\frac{3}{4}, \frac{6}{7}\right)\).

Recall the distance formula between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\): \[d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}\]

Calculate the difference in the \(x\)-coordinates: \[x_2 - x_1 = \frac{3}{4} - \left(-\frac{1}{4}\right) = \frac{3}{4} + \frac{1}{4}\]

Calculate the difference in the \(y\)-coordinates: \[y_2 - y_1 = \frac{6}{7} - \left(-\frac{1}{7}\right) = \frac{6}{7} + \frac{1}{7}\]

Substitute these differences into the distance formula, simplify the expression under the square root to get the simplified radical form, and then approximate the decimal value rounded to two decimal places.

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주요 개념

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Distance Formula

The distance formula calculates the length between two points in a coordinate plane. It is derived from the Pythagorean theorem and given by √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the straight-line distance between points with coordinates (x1, y1) and (x2, y2).

Simplifying Radical Expressions

Simplifying radicals involves reducing the square root expressions to their simplest form by factoring out perfect squares. This process makes the answer more exact and easier to interpret before rounding. For example, √50 can be simplified to 5√2.

Rounding Decimal Numbers

Rounding is the process of approximating a number to a specified number of decimal places for clarity and practicality. After calculating the exact distance, rounding to two decimal places provides a concise and usable numerical answer, such as 1.23 instead of 1.23456.