Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7, 2) and (x2, y2) = (1, −1). Find √[(x2 − x1)² + (y2 − y₁)²]. Express the - answer in simplified radical form.

Verified step by step guidance

1. Identify the formula for the distance between two points in a coordinate plane: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
2. Substitute the given coordinates (x₁, y₁) = (7, 2) and (x₂, y₂) = (1, −1) into the formula: d = √[(1 − 7)² + (−1 − 2)²].
3. Simplify the expressions inside the parentheses: (1 − 7) becomes −6, and (−1 − 2) becomes −3. The formula now becomes d = √[(-6)² + (-3)²].
4. Square the values: (-6)² = 36 and (-3)² = 9. The formula now becomes d = √[36 + 9].
5. Add the squared values: 36 + 9 = 45. The distance formula simplifies to d = √45. Simplify the radical by factoring: √45 = √(9 × 5) = 3√5.

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 √[(x2 - x1)² + (y2 - y1)²]. This formula calculates the straight-line distance between points (x1, y1) and (x2, y2), which is essential for solving problems involving spatial relationships.

Simplified Radical Form

Simplified radical form refers to the expression of 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. Understanding how to simplify radicals is crucial for presenting answers in a clear and concise manner, especially in algebraic contexts.

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 (x, y), which indicates its position relative to the axes. Familiarity with the coordinate system is vital for interpreting and solving problems involving points, distances, and geometric figures.