Problem 4 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(2,3) \text { and }(3,5)$$

Step-by-Step Solution

Answer

The distance between the points is \( \sqrt{5} \) units

Step 1: Identify the coordinates

The coordinates given are point P=(2,3) and point Q=(3,5). We denote them as P=(x1,y1) and Q=(x2,y2)

Step 2: Apply the Distance formula

The formula for distance between two points in a 2-dimensional plane is given by \(D = \sqrt{(x2-x1)^2 + (y2-y1)^2}\) . Inserting the provided values gives \(D = \sqrt{(3-2)^2 + (5-3)^2}\)

Step 3: Calculate

Perform the calculations \(\sqrt{(1)^2 + (2)^2} = \sqrt{1+4} = \sqrt{5}\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry, enabling the study of geometric shapes using a coordinate system. This system uses numbers to represent points on a plane. In most cases, we use the Cartesian coordinate system, which is a two-dimensional grid where any point can be identified by an ordered pair of numbers, \( (x, y) \)—the x-coordinate (horizontal position) and the y-coordinate (vertical position).

To visualize it, imagine a map where each location has a specific address—this is how coordinate geometry helps us navigate the mathematical 'landscape'. It's crucial for understanding shapes, sizes, and positions of figures and is widely used not only in mathematics, but also in fields like physics, engineering, and computer graphics.

Algebraic Concepts

Algebraic concepts form the foundation of many calculations within mathematics. They involve the use of symbols and letters to represent numbers and quantities in formulas and equations. This symbolic language helps us describe relationships, patterns, and changes. For instance, the distance formula \( D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \) is derived from the Pythagorean theorem, which is a fundamental concept in algebra dealing with the relationship between the sides of right-angled triangles.

Linking Geometry and Algebra

Through the distance formula, algebra connects directly to coordinate geometry by giving a method to calculate the distance between two points on a plane. By substituting specific coordinates into the formula, you can find the actual distance as you would measure with a ruler, but through pure computation.

Distance Between Two Points

The distance between two points is simply how far apart they are from each other in space. It's a straightforward concept that we deal with in everyday life, like when using a map or playing sports. In mathematics, specifically in coordinate geometry, calculating this distance involves combining both geometrical understanding and algebraic manipulation.

To find the distance between two points, we use the distance formula derived from the Pythagorean theorem. Imagine drawing a right-angled triangle using the two points as corners. The horizontal and vertical sides represent the differences in x and y coordinates, and the hypotenuse is the distance between the points. When you substitute the coordinates into the distance formula, you are essentially computing the length of this hypotenuse. Therefore, the seemingly simple act of measuring distance in math encapsulates a deep connection between geometric figures and algebraic expressions.