Problem 6 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,0) \text { and }(3,-4) $$

Step-by-Step Solution

Answer

The distance between points (0,0) and (3,-4) is 5 units.

Step 1: Identify the Coordinates

Identify the coordinates of the two points. The first point is (0,0) and has coordinates \(x_1=0\), \(y_1=0\). The second point is (3,-4) and has coordinates \(x_2=3\), \(y_2=-4\).

Step 2: Substitute into the Distance Formula

Substitute these coordinates into the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to get \(d = \sqrt{(3-0)^2 + (-4-0)^2}\). Enforce the subtraction inside the squares to get \(d = \sqrt{3^2 + -4^2}\).

Step 3: Solve for Distance

Solve the equation to get the distance. Square 3 and -4 to get \(d = \sqrt{9 + 16} = \sqrt{25}\). The square root of 25 is 5.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to represent geometric figures in a numerical format using pairs of numbers, known as coordinates. These coordinates define the position of points on a two-dimensional plane. The plane itself is determined by two perpendicular lines, or axes, usually called the x-axis and y-axis.

• The horizontal component of any point is referred to as its x-coordinate.
• The vertical component is referred to as its y-coordinate.

This way of describing points is known as the Cartesian coordinate system. With this system, any point’s location is given by an ordered pair \(x, y\), where x indicates the point's distance from the origin along the x-axis, and y indicates the distance along the y-axis.
For example, the given points (0,0) and (3,-4) tell us that:

• (0,0) is located at the origin, where both x and y equal zero.
• (3,-4) is 3 units to the right of the origin and 4 units down.

Understanding coordinate geometry is essential for solving many problems, such as finding the distance between these two points.

Distance Between Points

The distance between two points on a coordinate plane can be calculated using the distance formula. This formula is derived from the concept of a right triangle, where the distance represents the hypotenuse. The two other sides are the differences in the x and y coordinates respectively.
The formula used is:

• \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)

This equation breaks down into simpler steps:

• Subtract the x-coordinate of the first point from the x-coordinate of the second. This gives one side of the triangle.
• Subtract the y-coordinate of the first point from the y-coordinate of the second. This provides the other side of the triangle.
• Square both results, add them together, and take the square root to find the hypotenuse, or the distance.

Using the given points (0,0) and (3,-4), substituting into the formula helps illustrate this process: Calculate \( d = \sqrt{(3-0)^2 + (-4-0)^2} \). Make sure each step maintains the operations correctly to arrive at \( d = 5 \), showing the distance clearly.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, often used in conjunction with coordinate geometry to find distances. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This is expressed as:

• \( c^2 = a^2 + b^2 \)

In coordinate geometry, this idea is adapted to calculate the distance between two points, representing the hypotenuse in a right triangle formed by the vertical and horizontal distances between these points.
For example, consider the line that connects (0,0) and (3,-4). Here:

• One side (a) is the distance along the x-axis, which is \(3\).
• The other side (b) is the distance along the y-axis, which is \(4\).

Apply the theorem: \( c^2 = 3^2 + 4^2 = 9 + 16 = 25 \). Therefore, \( c = \sqrt{25} = 5 \).
This shows how the Pythagorean Theorem underlies the distance formula, supporting its accuracy in calculating distances in coordinate geometry.