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, -6) and (3, −4)

Verified step by step guidance

1. Identify the formula for the distance between two points in a coordinate plane: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the given points.
2. Substitute the coordinates of the points \((-2, -6)\) and \((3, -4)\) into the formula. Here, \(x_1 = -2\), \(y_1 = -6\), \(x_2 = 3\), and \(y_2 = -4\). The formula becomes \( d = \sqrt{(3 - (-2))^2 + ((-4) - (-6))^2} \).
3. Simplify the expressions inside the parentheses. \(3 - (-2)\) becomes \(3 + 2 = 5\), and \((-4) - (-6)\) becomes \(-4 + 6 = 2\). The formula now looks like \( d = \sqrt{5^2 + 2^2} \).
4. Square the values of \(5\) and \(2\). \(5^2 = 25\) and \(2^2 = 4\). The formula becomes \( d = \sqrt{25 + 4} \).
5. Add the values inside the square root. \(25 + 4 = 29\), so the formula becomes \( d = \sqrt{29} \). If necessary, simplify \(\sqrt{29}\) to its decimal form rounded to two decimal places.

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.

Coordinate System

A coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point in this system is represented by an ordered pair of numbers (x, y), indicating its position relative to the axes. Understanding how to plot points and interpret their coordinates is essential for applying the distance formula effectively.

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 concept is important when calculating distances, as it often results in a more concise and understandable answer before rounding to decimal places.