Problem 12 - College Algebra and Calculus: An Applied Approach

Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((1.451,3.051),(5.906,11.360)\)

Step-by-Step Solution

Answer

The plotted points are (1.451,3.051) and (5.906,11.360), the distance between the points is calculated as \[ d = \sqrt{{(5.906-1.451)}^2 + {(11.360-3.051)}^2} \] and the midpoint of the line segment joining these points is obtained as \[((1.451+5.906)/2 , (3.051+11.360)/2)\].

Step 1: Plotting the points

Given two points (1.451, 3.051) and (5.906, 11.360). Mark them on a two-dimensional plane where the x-coordinate is 1.451, 5.906 and y-coordinate is 3.051 and 11.360 respectively.

Step 2: Calculating the distance

To compute the distance between two points (x1, y1) and (x2, y2), we apply the formula \[d= \sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}\]. Substituting the given points (1.451, 3.051) and (5.906, 11.360) into the equation, we get \[ d = \sqrt{{(5.906-1.451)}^2 + {(11.360-3.051)}^2} \].

Step 3: Finding the midpoint

The formula for the midpoint of a line segment connecting two points (x1, y1) and (x2, y2) is \[((x1+x2)/2 , (y1+y2)/2)\]. Applying the provided points into the formula, we obtain the midpoint as \[((1.451+5.906)/2 , (3.051+11.360)/2)\].

Key Concepts

Distance Formula

The distance formula helps us calculate the linear distance between two points in a two-dimensional plane. In coordinate geometry, the two points are usually denoted as

• \((x_1, y_1)\) and \((x_2, y_2)\).

To find the distance, we use the equation:\[d = \sqrt{{(x_2-x_1)}^2 + {(y_2-y_1)}^2}.\]This formula is derived from the Pythagorean theorem, which explains it in terms of a right triangle.
The expression

• \((x_2-x_1)\) represents the horizontal (x-axis) distance,
• while \((y_2-y_1)\) gives the vertical (y-axis) distance.

Squaring these differences and taking their sum forms the hypotenuse squared (as in trigonometry).
Taking the square root gives the actual distance between the two points.
In practice, simply plug in the coordinates of the two points to quickly determine how far apart they are.

Coordinate Geometry

Coordinate geometry acts as a bridge between algebra and geometry. It helps us depict algebraic equations in geometric forms.
In this section, we use coordinates (x, y) to represent points on a plane.
This approach provides a handy way to solve geometric problems using algebra.In this specific problem, you are given two sets of coordinates:

• \((1.451, 3.051)\)
• \((5.906, 11.360)\).

By plotting these points on a graph, we can visualize concepts like distance and midpoint by analyzing these coordinate pairs.
Each point corresponds to a unique position on the two-dimensional plane.
By moving across x-coordinates (left to right) and up y-coordinates (up and down), coordinate geometry makes it easier to understand the spatial relationship between points.

Two-Dimensional Plane

A two-dimensional plane is a flat surface extending infinitely in two dimensions: length and width.
This concept is central to coordinate geometry, where the plane is represented by two number lines intersecting at right angles.
The intersection point of these number lines is known as the origin and is denoted as \((0, 0)\).On this plane, any point can be described using an ordered pair \((x, y)\).

• The first number, \(x\), represents the point's position along the horizontal axis (left-right).
• The second number, \(y\), represents the position along the vertical axis (up-down).

This system allows us to deal with geometric figures by plotting them on a graph and analyzing their properties.
The two-dimensional plane provides a foundation for all sorts of geometric explorations and problem-solving strategies.