Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-2, -6) and (3, −4)

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Identify the coordinates of the two points: Point 1 is \((-2, -6)\) and Point 2 is \((3, -4)\).

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

Substitute the given coordinates into the formula: \[d = \sqrt{(3 - (-2))^2 + (-4 - (-6))^2}\]

Simplify inside the parentheses: \[d = \sqrt{(3 + 2)^2 + (-4 + 6)^2}\] which becomes \[d = \sqrt{5^2 + 2^2}\]

Calculate the squares and write the expression under the square root: \[d = \sqrt{25 + 4}\] Then simplify the radical if possible, and finally round the result 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 the coordinate plane. It is derived from the Pythagorean theorem and is given by √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the straight-line distance between points (x1, y1) and (x2, y2).

Simplifying Radical Expressions

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. This makes the answer more exact and easier to interpret before rounding. For example, √50 can be simplified to 5√2 by factoring 50 into 25 × 2.

Rounding Decimal Numbers

Rounding is the process of approximating a number to a specified number of decimal places for clarity or practicality. In this problem, after simplifying the radical, the distance should be rounded to two decimal places to provide a clear, usable numerical answer.