Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (2, 3) and (14, 8)

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Identify the coordinates of the two points: Point 1 is (2, 3) and Point 2 is (14, 8).

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: \[d = \sqrt{(14 - 2)^2 + (8 - 3)^2}\]

Simplify inside the square root by calculating the differences and then squaring them: \[d = \sqrt{(12)^2 + (5)^2}\]

Express the sum inside the square root and simplify if possible: \[d = \sqrt{144 + 25} = \sqrt{169}\] Then, find the simplified radical form and prepare to round to two decimal places if needed.

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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 distance between two points in the coordinate plane using their coordinates. 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.

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 work with before converting to a decimal approximation.

Rounding to Decimal Places

Rounding involves approximating a number to a specified number of decimal places for simplicity and clarity. In this problem, the final distance should be rounded to two decimal places to provide a clear, concise numerical answer.