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. (4, -1) and (-6, 3)

Verified step by step guidance

1. Start by identifying the coordinates of the two points: (x_1, y_1) = (4, -1) and (x_2, y_2) = (-6, 3).
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{((-6) - 4)^2 + (3 - (-1))^2}.
4. Simplify the expressions inside the square root: d = \sqrt{(-10)^2 + (4)^2}.
5. Calculate the squares and add them: d = \sqrt{100 + 16}.

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, √8 can be simplified to 2√2. This form is often preferred in mathematics for clarity and ease of further calculations, especially when dealing with distances or geometric problems.

Rounding Numbers

Rounding numbers is the process of adjusting a number to a specified degree of accuracy, often to make it easier to work with or understand. In this context, rounding to two decimal places means adjusting the calculated distance to the nearest hundredth. This is important in practical applications where exact values are less critical than a reasonable approximation.