Problem 50 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((2,-3),(4,-8)\)

Step-by-Step Solution

Answer

The exact distance is \(\sqrt{29}\).

Step 1: Compute the Horizontal Distance

Identify the x-coordinates of the two points: the first point x-coordinate is 2 and the second point x-coordinate is 4. Calculate the horizontal distance by subtracting the x-coordinate of the first point from that of the second point: \[ x_2 - x_1 = 4 - 2 = 2 \].

Step 2: Compute the Vertical Distance

Identify the y-coordinates of the two points: the first point y-coordinate is -3 and the second point y-coordinate is -8. Calculate the vertical distance by subtracting the y-coordinate of the first point from that of the second point:\[ y_2 - y_1 = -8 - (-3) = -8 + 3 = -5 \].

Step 3: Apply the Distance Formula

Use the distance formula to calculate the distance between the two points: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].Substitute the computed horizontal and vertical distances:\[ d = \sqrt{(2)^2 + (-5)^2} = \sqrt{4 + 25} \].

Step 4: Calculate the Exact Distance

Simplify the expression inside the square root and then compute the exact distance:\[ d = \sqrt{29} \].Thus, the exact distance between the points \((2, -3)\) and \((4, -8)\) is \(\sqrt{29}\).

Key Concepts

Horizontal Distance

When you think about horizontal distance, imagine traveling along a straight path that runs from left to right across your screen, much like driving along a straight road.
For points located on a coordinate plane, the horizontal distance is found by focusing on the x-coordinates. Recall that the x-coordinate tells you how far left or right a point is on the plane.
To calculate it, subtract the x-coordinate of the starting point from the x-coordinate of the ending point.
Here's a simple guideline:

• First, identify the x-coordinates from your pair of points. For example, for the points \( (2, -3) \) and \( (4, -8) \), these are 2 and 4, respectively.
• Subtract the first x-coordinate from the second: \( x_2 - x_1 = 4 - 2 = 2 \).

This tells you that, horizontally, the points are 2 units apart.By understanding this, you know how far left or right one point is from another.

Vertical Distance

Vertical distance takes you on a journey up and down the graph, akin to an elevator ride moving between floors in a building.
By focusing on the y-coordinates of your points, you figure out how far apart the points are vertically. The y-coordinate indicates the height of a point on the graph.
To find the vertical distance, you'll calculate the difference between the y-coordinates of the two points:

• Take the y-coordinates from your points, like in \( (2, -3) \) and \( (4, -8) \), which are -3 and -8.
• Subtract the first y-coordinate from the second, by being careful with signs: \[ y_2 - y_1 = -8 - (-3) = -8 + 3 = -5 \]

The negative result means you’re looking at the distance in a direction towards a less positive or more negative y value.
The concept of vertical distance is crucial in visualizing how high or low points appear relative to each other.

Coordinate Plane

The coordinate plane is like a giant map that helps you navigate the position of points with precision. It is made up of two intersecting lines known as the x-axis (horizontal line) and the y-axis (vertical line). These lines divide the plane into four quadrants, giving you a clear grid to place points.
Each point on this plane is identified by a pair of numbers, its coordinates. The first number in any coordinates pair refers to its position on the x-axis, and the second refers to the y-axis.
For example, in the point \( (2,-3) \), 2 tells us how far along the x-axis the point is, and -3 tells us its position along the y-axis.
Navigating a coordinate plane calls for:

• Understanding that each point is marked with an \( (x, y) \) position.
• Intersections of the axes form a center point called the origin, at \( (0, 0) \).
• Using both axes' positions to describe the point's exact location.

Thus, any problem you solve on the coordinate plane makes use of these simple yet organized grid structures that allow distance measurements, beyond just horizontal or vertical, by applying the distance formula to find the direct path between any two points.