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

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

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: \[d = \sqrt{(-6 - 4)^2 + (3 - (-1))^2}\]

Simplify inside the parentheses: \[d = \sqrt{(-10)^2 + (4)^2}\]

Calculate the squares and express the distance in simplified radical form: \[d = \sqrt{100 + 16} = \sqrt{116}\]. Then, simplify the radical if possible before rounding to two decimal places.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the square root of the sum of the squares of the differences in x-coordinates and y-coordinates: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Simplified Radical Form

Simplified radical form means expressing a square root in its simplest form by factoring out perfect squares. This makes the expression easier to interpret and use in further calculations, such as \( \sqrt{50} = 5\sqrt{2} \).

Rounding to Decimal Places

Rounding involves approximating a number to a specified number of decimal places for clarity or practical use. Here, after simplifying the radical, the distance should be rounded to two decimal places, ensuring the answer is both precise and easy to read.

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