In Exercises 1–18, 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)

Verified step by step guidance

1. Identify the coordinates of the two points: Point 1 is \((x_1, y_1) = (2, 3)\) and Point 2 is \((x_2, y_2) = (14, 8)\).
2. Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
3. Substitute the coordinates into the formula: \(d = \sqrt{(14 - 2)^2 + (8 - 3)^2}\).
4. Simplify the expressions inside the square root: \(d = \sqrt{12^2 + 5^2}\).
5. Calculate the squares and add them: \(d = \sqrt{144 + 25}\), then simplify the square root if possible.

Key Concepts

Distance Formula

The distance formula is a mathematical equation used to determine the distance between two points in a Cartesian coordinate system. It is derived from the Pythagorean theorem and is expressed as d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points. This formula allows for the calculation of the straight-line distance between any two points.

Simplified Radical Form

Simplified radical form refers to expressing a square root in its simplest terms, where no perfect square factors remain under the radical sign. For example, √18 can be simplified to 3√2. This form is often preferred in mathematics for clarity and ease of understanding, especially when dealing with distances or lengths.

Rounding Numbers

Rounding numbers is the process of adjusting a number to a specified degree of accuracy, typically to make it simpler or easier to work with. In this context, rounding to two decimal places means adjusting the calculated distance to the nearest hundredth. This is important for providing a clear and concise answer, especially in practical applications where exact values may not be necessary.