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Chapter 35: Problem 626

Find the distance between the point \(\mathrm{A}(1,3)\) and \(\mathrm{B}(5,3)\).

Short Answer

Expert verified
The distance between point A(1,3) and point B(5,3) is 4 units, since their y-coordinates are the same and the difference between their x-coordinates is |5 - 1| = 4.

Step by step solution

01

Identify the given points

We have been given two points, point A(1,3) and point B(5,3).
02

Check if the y-coordinates are the same

Observe that the points A and B have the same y-coordinates: the y-coordinate of point A is 3, and the y-coordinate of point B is also 3.
03

Calculate the distance between the x-coordinates

Since the y-coordinates are the same, our task is simplified. We can find the distance between A and B by simply calculating the difference between their x-coordinates: Distance = |x2 - x1| where x1 is the x-coordinate of point A (1) and x2 is the x-coordinate of point B (5).
04

Substitute the values and find the distance

Now, replace x1 and x2 with their respective values and find the distance: Distance = |5 - 1| Distance = |4| Distance = 4 The distance between point A(1,3) and point B(5,3) is 4 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 a branch of mathematics that deals with defining and representing shapes and figures in a coordinate plane using a set of numbers. This field enables mathematicians and students alike to describe the exact location of points using coordinates. Here's why coordinate geometry is significant:
  • It allows for the visual representation of equations and inequalities as graphs.
  • It provides tools such as the Distance Formula and Midpoint Formula to calculate distances and midpoints between points.
  • Coordinate geometry is pivotal in analyzing geometrical shapes like lines, curves, and angles and understanding their properties using algebra.
In our example exercise, we applied a principle of coordinate geometry to determine the distance between two points on a plane. This is achieved primarily through the understanding of coordinates and their operation in the Cartesian system.
Cartesian Plane
The Cartesian Plane, often known as the coordinate plane, is a two-dimensional plane determined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Here's what makes the Cartesian Plane so essential:
  • It allows for the plotting of points using pairs of numerical coordinates. Each point corresponds to a location defined by an x-coordinate and a y-coordinate.
  • The intersection point of these axes is called the origin, represented as (0,0), where both values are zero.
  • This system is crucial for plotting and analyzing algebraic equations and their graphs, allowing a visual approach to solving equations.
In the exercise, we effectively used the Cartesian Plane to identify the coordinates of points A and B. These coordinates helped us leverage the Distance Formula to calculate the distance between them.
Points on a Line
Points on a line in a Cartesian Plane suggest a linear relationship between the coordinates of those points. Here's why understanding this concept is important:
  • If two or more points lie on the same horizontal or vertical line, certain distances or properties can be calculated more easily.
  • Points on a line can be used to determine the slope of the line, showing us the rate of change or steepness.
  • This concept is fundamental in understanding collinearity, where three or more points lie on the same straight line.
In the provided exercise, we recognized that both points A (1,3) and B (5,3) lie on the same horizontal line because their y-coordinates are identical. This property simplified our calculation, as we could directly find the distance using just their x-coordinates.

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Most popular questions from this chapter

Describe the location of the point \((-1,-2)\) and then plot it on coordinate graph paper.

Find the coordinates of the midpoints of \(\mathrm{AC}\) and \(\mathrm{BC}\), of triangle \(\mathrm{ABC}\), with coordinates \(\mathrm{A}(6,7), \mathrm{B}(-11,0), \mathrm{C}(1,-5)\).

Find the coordinates of the midpoint of the line segment which joins the point \(R(4,6)\) and the point \(S(8,-2)\). (See figure)

Find the distance between the points on the real number line whose coordinates are \(-4\) and 3 .

Find the distance between the point \(\mathrm{C}(-3,-2)\) and the point \(\mathrm{D}(-3,4)\)
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