Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. \(A(5,0), B(12,0)\)

Short answer

The distance between points A(5,0) and B(12,0) is 7 units.

Step-by-step solution

Understand the Distance Formula in Two Dimensions

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) in the Cartesian plane is calculated using the distance formula: \(d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\). This formula is derived from the Pythagorean theorem to find the hypotenuse of a right triangle.

Identify the Provided Points

The given points are \(A(5,0)\) and \(B(12,0)\). Here, \(x_1 = 5\), \(y_1 = 0\), \(x_2 = 12\), and \(y_2 = 0\). Since both points share the same y-coordinate, they lie on the horizontal line y = 0.

Apply the Distance Formula

Substitute the values into the distance formula: \(d = \sqrt{{(12 - 5)}^2 + {(0 - 0)}^2}\). This simplifies to \(d = \sqrt{{7}^2}\), as the y-coordinates are the same.

Calculate the Distance

Simplify the expression to find the distance: \(d = \sqrt{49} = 7\). Since there is no need for further calculations, the final distance is 7 units.

Key concepts

Cartesian Plane

The Cartesian Plane is a two-dimensional plane defined by two perpendicular lines, usually called the x-axis and y-axis. This plane allows us to define the location of any point using coordinates. Each point is identified by a pair of numbers written as \(x, y\). The first number represents the position along the horizontal axis (the x-axis), and the second number represents the position along the vertical axis (the y-axis).

• The Cartesian Plane is used to plot graphs, solve geometry problems, and find distances between points.
• The origin is the point where the axes intersect, labeled as \(0, 0\).
• Each axis is marked with equal units of distance, and movement to the right or up increases the positive value of the coordinates.

Understanding how to navigate and plot points on this plane is crucial as it forms the foundation for geometry and algebra in mathematics.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It relates the lengths of the sides of a right triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse.

• This theorem is the backbone of the Distance Formula used in a Cartesian Plane to calculate the distance between two points.
• By treating the distance between two points as the hypotenuse of a right triangle, we use the theorem to find this distance.
• For points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the horizontal leg is \((x_2 - x_1)\) and the vertical leg is \((y_2 - y_1)\).

This conversion from point coordinates into triangle sides is what allows us to calculate an exact straight-line distance using algebraic methods.

Points on a Line

When points share the same y-coordinate, like \(A(5, 0)\) and \(B(12, 0)\), they lie on a horizontal line. In this scenario, the calculation of distance simplifies significantly. Here's why:

• If two points are on a horizontal line, you only need to find the difference in their x-coordinates.
• The distance between such points is a simple arithmetic calculation using just \(|x_2 - x_1|\).
• In our example, since y-coordinates are the same, the distance simply is \(12 - 5 = 7\).

Points on a straight line simplify many calculations in geometry, and recognizing these scenarios can save you time when working on problems involving distances and coordinates.