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Chapter 9: Problem 4

Find the distance between the points (5,-1) and (5,-8)

Short Answer

Expert verified
The distance between the points is 7 units.

Step by step solution

01

Identify the Coordinates

Firstly, identify and list the coordinates. Here, they are (5, -1) and (5, -8)
02

Calculate the Difference in Y-Coordinates

Since the x-coordinates of the two points are equal, the distance between them is the absolute difference between the y-coordinates. So the distance would be |-1 - (-8)| = |7| = 7 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate Geometry is an essential tool in mathematics. It involves using a coordinate plane, which is defined by an x-axis and a y-axis, to describe the positions of points. Each point on this plane has a unique set of coordinates, given as \(x, y\), where \(x\) is the horizontal position and \(y\) is the vertical position. This system allows us to visualize and solve geometric problems using algebra.
  • Every point in the coordinate plane is described by a pair of numbers.
  • These numbers, called the coordinates, tell us exactly where the point is.
  • The first number is the x-coordinate: it tells us the horizontal shift from the origin.
  • The second number is the y-coordinate: it shows the vertical shift from the origin.
Understanding coordinate geometry helps in calculating distances between points, finding midpoints, and working with equations of lines and curves.
Distance Calculation
When calculating the distance between two points in coordinate geometry, we can use the Distance Formula. However, when two points share the same x-coordinate or y-coordinate, the calculation simplifies.For two points, such as \( (x_1, y_1) \) and \( (x_2, y_2) \), the Distance Formula is generally:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In this scenario, our points are (5, -1) and (5, -8), sharing the x-coordinate 5. Therefore, we only need to measure the difference in their y-coordinates, bypassing the use of the formula.
  • Identify when coordinates share an axis value 鈥� this simplifies your task.
  • Use absolute differences for direct calculations, ensuring we measure positive distance.
  • The simplification can be a huge time-saver!
By focusing on shared coordinates, we efficiently find the vertical or horizontal distance between points.
Vertical Distance
Vertical Distance is the measure of how far a point moves along the y-axis within a coordinate plane. This type of calculation arises when two points share the same x-coordinate.
  • Identify identical x-coordinates: this indicates vertical measurement.
  • Calculate the distance with the formula: \( |y_2 - y_1| \)
  • This results in the absolute difference between the two y-coordinates.
In our example, using points (5, -1) and (5, -8), we calculate \( |-1 - (-8)| = |7| = 7 \). Thus, the vertical distance is 7 units.Vertical distances are practical in real-life applications, such as measuring heights or depth differences. By understanding this calculation, you can easily measure vertical changes irrespective of the range or points involved.

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Most popular questions from this chapter

Determine the equations in standard form of two different hyperbolas that satisfy the given conditions. Center at (0,0)\(;\) transverse axis of length \(12 ;\) slope of one asymptote is 4

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