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Chapter 3: Problem 90

Find the midpoint between the given two points. (0,0) and (1,1)

Short Answer

Expert verified
The midpoint is \( \left( \frac{1}{2}, \frac{1}{2} \right) \).

Step by step solution

01

Identify the Formula for Midpoint

The midpoint formula for a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
02

Substitute the Coordinates

Given points are \((0,0)\) and \((1,1)\). Substitute these into the midpoint formula:\[ \text{Midpoint} = \left( \frac{0 + 1}{2}, \frac{0 + 1}{2} \right) \]
03

Calculate the Midpoint

Calculate the values by performing the arithmetic operations:\[ \text{Midpoint} = \left( \frac{1}{2}, \frac{1}{2} \right) \]
04

Verify the Solution

The computed midpoint \( \left( \frac{1}{2}, \frac{1}{2} \right) \) is equidistant from both points \((0,0)\) and \((1,1)\), confirming that the solution is correct.

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Key Concepts

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

Coordinate Plane
The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates reflect a point's position in relation to two intersecting perpendicular lines called axes. One of these is the x-axis (horizontal), and the other is the y-axis (vertical).
  • The coordinate plane allows for plotting points, lines, and curves through numeric representation.
  • Every point on this plane can be identified by an ordered pair \( (x, y) \).
The origin, denoted as \( (0, 0) \), is where the x-axis and y-axis intersect.
In our exercise, the midpoint was calculated on the coordinate plane using coordinates (0,0) and (1,1). This suggests a linear segment joining these two points.
Arithmetic Operations
Arithmetic operations are foundational mathematical processes such as addition, subtraction, multiplication, and division. These operations are key in manipulating numbers to solve various types of problems.
  • Addition combines two numbers to get a sum.
  • Division splits a number into equal parts.
In the context of the midpoint formula, both addition and division operations are utilized.
For example, when finding the midpoint between \( (0,0) \) and \( (1,1) \), you add the x-coordinates and y-coordinates separately: \[ \frac{0 + 1}{2} \]This operation gives us the average of both x-coordinates and y-coordinates, thus determining the center point of the line segment joining the two coordinates.
Equidistant
The term equidistant refers to being at equal distances from two or more points. In geometry, finding a point that is equidistant from two other points is essential for accurately determining midpoints, centers, and symmetry.
The midpoint, calculated as \( \left( \frac{1}{2}, \frac{1}{2} \right) \), is equidistant from both \( (0,0) \) and \( (1,1) \).
This means that the distance from the midpoint to each of the original points is precisely the same, confirming the midpoint calculation was correctly executed.
  • Being equidistant validates the balance and divides the segment into two equal halves.
  • Visualizing this concept on the coordinate plane makes it clear and intuitive.
Understanding equidistance helps reinforce logical thinking about space and symmetry in mathematics.

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