Problem 43 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((0,0),(3,-4)\)

Step-by-Step Solution

Answer

The exact distance between the points is 5 units.

Step 1: Recognize the Distance Formula

The distance formula is used to find the distance between two points in a Cartesian plane. It is derived from the Pythagorean theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Identify the Coordinates of the Points

The given points are \((0, 0)\) and \((3, -4)\). Let \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (3, -4)\). Identify these values for substitution into the distance formula.

Step 3: Substitute the Coordinates into the Distance Formula

Substitute \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 3\), and \(y_2 = -4\) into the distance formula:\[ d = \sqrt{(3 - 0)^2 + (-4 - 0)^2} \]

Step 4: Simplify Inside the Square Root

Calculate the differences: \((3 - 0) = 3\) and \((-4 - 0) = -4\). Now, substitute these differences back into the formula:\[ d = \sqrt{3^2 + (-4)^2} \]

Step 5: Calculate the Squares

Calculate the square of each difference: \(3^2 = 9\) and \((-4)^2 = 16\). Substitute these squared values back into the equation:\[ d = \sqrt{9 + 16} \]

Step 6: Final Calculation of the Square Root

Add the squared terms together:\[ 9 + 16 = 25 \]Then, take the square root of the sum:\[ d = \sqrt{25} = 5 \]

Key Concepts

Cartesian Plane

The Cartesian plane is a two-dimensional surface defined by two perpendicular axes. These axes are called the x-axis and the y-axis. It is a fundamental concept in geometry, as it is used to plot points, lines, and curves, and to analyze their relationships in a flat surface.
The point where the x-axis and y-axis intersect is known as the origin, represented by the coordinates (0, 0).
Each location on the plane is specified by an ordered pair of numbers, which are known as coordinates.

• The horizontal axis is the x-axis.
• The vertical axis is the y-axis.

Understanding the Cartesian plane is essential for calculating the distance between two points using their coordinates.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

This theorem provides the foundation for deriving the distance formula used in a Cartesian plane.
By considering the two coordinates (0, 0) and (3, -4) as forming a right triangle with the x-axis and y-axis, you can visualize the calculation of the distance as finding the hypotenuse.
So, when you apply the Pythagorean theorem to these two points, you are essentially finding the straight-line distance, or hypotenuse, between them.

• The formula for the Pythagorean theorem is: \( c^2 = a^2 + b^2 \).
• In the context of coordinates, this transforms to the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Coordinates

Coordinates are used to define the position of a point on the Cartesian plane. Each point has an (x, y) pair, where 'x' represents the horizontal position and 'y' represents the vertical position.

• The first number in the pair is the x-coordinate.
• The second number is the y-coordinate.

From the exercise, the given points were (0, 0) and (3, -4).

Using these coordinates, you can determine the distance between the two points using the distance formula.
The difference in the x-values ( 3 - 0) and the difference in the y-values ( -4 - 0) were used in the formula to calculate the distance. By accurately working with coordinates, it becomes straightforward to measure the direct distance between any two points on the plane.