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Chapter 1: Problem 37

Find the coordinates of the midpoint of a segment having the given endpoints. $$A(8,4), B(12,2)$$

Short Answer

Expert verified
The midpoint of the segment with endpoints \(A(8,4)\) and \(B(12,2)\) is \(M(10,3)\).

Step by step solution

01

Identify the formula

The formula to find the midpoint M of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
02

Substitute the coordinates

Now, substitute the coordinates of the endpoints into the midpoint formula:- \( x_1 = 8 \), \( y_1 = 4 \)- \( x_2 = 12 \), \( y_2 = 2 \)Thus, the formula becomes:\[ M = \left( \frac{8 + 12}{2}, \frac{4 + 2}{2} \right) \]
03

Calculate the sums

First, calculate the sum for the x-coordinates: \( x_1 + x_2 = 8 + 12 = 20 \).Then, calculate the sum for the y-coordinates:\( y_1 + y_2 = 4 + 2 = 6 \).
04

Divide by 2

Continue by dividing each sum by 2 to find the coordinates of the midpoint:For the x-coordinate: \( \frac{20}{2} = 10 \).For the y-coordinate: \( \frac{6}{2} = 3 \).
05

Write the midpoint coordinates

Combine the results to determine the coordinates of the midpoint: \( M(10, 3) \).

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Key Concepts

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

Understanding Coordinates
Coordinates are pairs of numbers that describe the position of a point in a two-dimensional space. In geometry, the coordinate system is usually based on the Cartesian plane, which consists of two perpendicular number lines that intersect at a point called the origin.
  • The pair is always written in the form \( (x, y) \), where \(x\) is the horizontal position, and \(y\) is the vertical position.
  • Coordinates allow us to pinpoint the precise location of a point on the plane.
For example, the point \( (8, 4) \) tells us the point is 8 units along the x-axis and 4 units along the y-axis. Flipping these around, to \( (4, 8) \), would place the point a very different location, showcasing the importance of order.
Endpoints Explained
Endpoints are the two points that form the ends of a line segment. When you imagine a line segment, picture it as a straight path connecting two endpoints. These two points will have their own coordinates that define their positions on the Cartesian plane.
  • In our problem, the endpoints are \( A(8,4) \) and \( B(12,2) \).
  • The endpoints are critical for determining the midpoint of the segment.
By knowing the coordinates of the endpoints, we gain full control over the geometric properties of the line segment they form, such as its length and midpoint.
Getting to Know the Line Segment
A line segment is part of a line that is bounded by two distinct endpoints. Unlike lines, segments do not stretch out infinitely. They have a specific start and end point, which makes them convenient for geometric calculations.
  • Each line segment in a graph is made up by connecting two endpoints with a straight line.
  • The segment forms a finite length, which can be calculated as the distance between the two endpoints.
The line segment formed by the points \( A \) and \( B \) in this case goes from \( (8,4) \) to \( (12,2) \). These endpoints enable us to apply the midpoint formula effectively. Segments help us in breaking down complex figures into manageable, measurable portions.
The Concept of the X-Coordinate
The x-coordinate is part of the coordinate pair that indicates the horizontal position of a point within the Cartesian plane. It is the first number in a pair \( (x, y) \) and tells us how far the point is from the y-axis.
  • In the midpoint formula, we add the x-coordinates and then divide by 2.
  • This gives us the x-coordinate of the midpoint, which tells us the central horizontal position of the line segment.
In our problem, the x-coordinates are 8 and 12. Adding them gives 20, and dividing by 2 provides the midpoint x-coordinate, which is 10. This is vital for ensuring the calculated midpoint is centered between the two endpoints on the x-axis.
What is the Y-Coordinate?
The y-coordinate determines the vertical position of a point in a two-dimensional plane. It is the second number in the coordinate pair \( (x, y) \), reflecting the distance above or below the x-axis.
  • To find the y-coordinate of the midpoint, sum the y-coordinates of the endpoints and divide by 2.
  • This calculation results in the midpoint's vertical position along the y-axis.
For our exercise, the given y-coordinates are 4 and 2, which sum to 6. Dividing by 2 results in a y-coordinate of 3. This completes the midpoint calculation, placing it accurately in the center vertically between the original endpoints on the graph.

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Most popular questions from this chapter

Martin makes pewter figurines. When a solid object with a volume of 1 cubic centimeter is submerged in water, the water level rises 1 milliliter. Martin pours \(200 \mathrm{mL}\) of water into a cup, submerges a figurine in it, and watches it rise to \(343 \mathrm{mL}\). What is the maximum amount of molten pewter needed to make a figurine? Explain.

Rays \(P Q\) and \(Q R\) are perpendicular. Point \(S\) lies in the interior of \(\angle P Q R .\) If \(m \angle P Q S=4+7 a \text { and } m \angle S Q R=9+4 a, \text { find } m \angle P Q S \text { and } m \angle S Q R\)

Determine whether each statement is sometimes, always, or never true. If two angles are supplementary and one is acute, the other is obtuse.

Find the value of the variable and \(Q R\), if \(Q\) is between \(P\) and \(R\). $$P R=7 n+8, P Q=4 n-3, Q R=6 n+2$$

OPEN ENDED Fold a sheet of paper. Open the paper and fold it again in a different way. Open the paper and label the geometric figures you observe. Describe the figures.
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