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Chapter 1: Problem 19

Find the midpoint of the line segment with the following endpoints. $$(1,-1),(5,5)$$

Short Answer

Expert verified
The midpoint of the line segment with endpoints \((1,-1)\) and \((5,5)\) is \((3, 2)\).

Step by step solution

01

Identify the Coordinates of the Endpoints

The given endpoints are \((1,-1)\) and \((5,5)\).So we have \(x1 = 1\), \(y1 = -1\), \(x2 = 5\), and \(y2 = 5\).
02

Apply the Midpoint Formula

Substitute the coordinates of the endpoints into the midpoint formula. Therefore, \((x1 + x2)/2 = (1 + 5)/2 = 3\) and \((y1 + y2)/2 = (-1 + 5)/2 = 2\).
03

Answer

So the midpoint of the line segment AB is at \((3, 2)\).

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Key Concepts

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

Line Segment
In geometry, a line segment is a part of a line that is bounded by two distinct endpoints. Imagine it as a straight path connecting two points in space. Unlike a full line, a line segment has a definite beginning and end. Visualization helps greatly in understanding this concept, so consider drawing or visualizing line segment AB with points A and B as its endpoints. This is the foundation for finding the midpoint, as the midpoint essentially divides this line segment into two equal parts. Understanding line segments is key in various mathematical applications, from basic geometry to advanced studies.
Coordinates
Coordinates are a system of numbers used to specify the position of points. Typically, they are in pairs like \((x,y)\) on a Cartesian plane. The x-coordinate describes a point's horizontal position, while the y-coordinate describes its vertical position. For the given exercise, the coordinates of the endpoints are \(1, -1\) and \(5, 5\). Learning to read and place coordinates on a graph enhances spatial reasoning and helps comprehend how different points relate to one another. Think of coordinates as precise addresses for locating points in a two-dimensional space.
Endpoints
Endpoints are the two points at which a line segment begins and ends. They are pivotal when using formulas, such as the midpoint formula, because they provide the specific coordinates needed for calculations. In this exercise, the endpoints given are \(1, -1\) and \(5, 5\). These endpoints provide the necessary information to calculate the midpoint of the line segment. Remember, endpoints help in determining the segment's length and its position in a coordinate plane.
Mathematics Education
Incorporating practical exercises in mathematics education helps in grasping abstract concepts. Using real-life examples or visual aids can turn a seemingly abstract topic into a tangible activity. The midpoint formula, used in the step-by-step solution, is essential in solving geometric problems and understanding spatial relationships. To support learning:
  • Use interactive tools like graphing software to visualize coordinates and line segments.
  • Simplify complex concepts by breaking problems into smaller steps.
  • Encourage the practice of related exercises to reinforce understanding.
These methods can enhance comprehension and make mathematics a more approachable subject for learners.

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Most popular questions from this chapter

Use interval notation to express the solution set of each inequality. $$|x|>2$$

The number of diagonals \(D\) of a polygon with \(n\) sides is given by the formula $$D=\frac{n}{2}(n-3)$$ a. Determine the number of sides of a polygon with 464 diagonals. b. Can a polygon have 12 diagonals? Explain.

Use a graphing utility. Graph: \(f(x)=\left|x^{2}-2 x\right|-3\)

If \(y=x^{2}-3 x+2,\) find \(x\) when \(y=0 .[1.1]\)

Solve by completing the square or by using the quadratic formula. $$\sqrt{2} x^{2}+3 x+\sqrt{2}-0$$
See all solutions

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