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Chapter 1: Problem 28

Find the distance between the points. $$ (1,4),(8,4) $$

Short Answer

Expert verified
The distance between (1,4) and (8,4) is 7 units.

Step by step solution

01

Identify coordinates

Firstly, recognize the coordinate points given. Here Point A is (1,4) and Point B is (8,4).
02

Apply the distance formula

Normally, the distance formula in 2D is given by \( \sqrt{(x_2 - x_1 )^2 + (y_2 - y_1 )^2} \). But since the y-coordinates of the points are equal (4-4=0), this becomes \( \sqrt{(x_2 - x_1 )^2} \).
03

Calculate the distance

Now, just plug the coordinates into the formula: \( \sqrt{(8 - 1 )^2} \), which simplifies to \( \sqrt{7^2} \). Here, \(7^2\) becomes 49 and the square root of 49 is 7.

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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 helps us understand geometric problems using a system of coordinates. This system allows us to locate points, lines, and shapes on a plane of cartesian coordinates, usually represented by an X and Y axis.

In this approach, each point is defined by a pair of numbers known as coordinates. For example, if a point is located at (1, 4), this means it is 1 unit across the X-axis and 4 units up the Y-axis. Coordinate Geometry is widely used to solve geometric problems using algebraic techniques. It allows for precise mathematical expressions and computations, making it fundamental in solving real-world problems efficiently.

  • Transforms geometric shapes into algebraic equations.
  • Facilitates intersection calculations of lines and shapes.
  • Enables calculation of distances between points directly.
Two-Dimensional Space
Two-Dimensional Space is a mathematical concept that includes only two dimensions: length and width. Unlike three-dimensional space, it lacks depth, making it simpler for calculations and analyses.

Understanding 2D space is crucial for solving problems in Coordinate Geometry. It involves working with a plane, where positions are identified by pairwise combinations of coordinates, such as (x, y).

Here's why Two-Dimensional Space is essential for many mathematical problems:
  • Simplifies complex calculations by reducing dimensions.
  • Allows us to visualize shapes and positions on a flat plane easily.
  • Helps in understanding relationships between shapes, lines, and points.
In 2D space, we work with graphs, plots, and diagrams to better understand spatial relationships and distances.
Mathematical Problem Solving
Mathematical Problem Solving involves a logical approach to find solutions to queries using mathematical methods and formulas. It's the backbone of efficient reasoning and calculation, essential for everyday problems and academic exercises alike.

Solving a problem like finding the distance between two points needs a clear strategy. Here's how you can tackle such problems:
  • Identify Given Data: Recognize coordinates or figures given in the problem. For instance, the points (1, 4) and (8, 4) tell us exact locations.
  • Choose the Right Formula: Use mathematical formulas that best fit the problem. In distance-related problems, the Distance Formula is apt.
  • Calculate Effectively: Substitute the values into the formula accurately. Perform simplification steps carefully to avoid errors.
  • Verify Results: Recheck calculations and ensure the result makes logical sense.
This structured approach helps enhance clarity, precision, and confidence in mathematical problem-solving.

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Most popular questions from this chapter

Find a mathematical model for the verbal statement. The gravitational attraction \(F\) between two objects of masses \(m_{1}\) and \(m_{2}\) is proportional to the product of the masses and inversely proportional to the square of the distance \(r\) between the objects.

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\left\\{\begin{array}{ll} -x, & x \leq 0 \\ x^{2}-3 x, & x>0 \end{array}\right. $$

In direct variation models of the form \(y=k x, k\) is called the _____ of _____ .

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt[3]{x-1} $$

Write a sentence using the variation terminology of this section to describe the formula. Surface area of a sphere: \(S=4 \pi r^{2}\)
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