Finding the Midpoint of a Segment with Given Endpoints

Oct 13, 2025 by windailys.com 110 views

Introduction

In mathematics, the midpoint of a line segment is the point that divides the segment into two equal parts. It is an essential concept in geometry and is used to find the midpoint of a line segment with given endpoints. In this article, we will learn how to find the midpoint of a segment with the given endpoints ${(0, -4)}$ and ${(9, 10)}$ .

What is a Midpoint?

A midpoint is a point that lies on a line segment and divides it into two equal parts. It is the average of the x-coordinates and the y-coordinates of the endpoints of the segment. The midpoint formula is used to find the midpoint of a line segment with given endpoints.

Midpoint Formula

The midpoint formula is given by:

${(x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$

where ${(x_1, y_1)}$ and ${(x_2, y_2)}$ are the given endpoints of the segment.

Finding the Midpoint of a Segment with Given Endpoints

To find the midpoint of a segment with given endpoints ${(0, -4)}$ and ${(9, 10)}$ , we can use the midpoint formula.

Step 1: Identify the Endpoints

The given endpoints are ${(0, -4)}$ and ${(9, 10)}$ .

Step 2: Apply the Midpoint Formula

We will apply the midpoint formula to find the midpoint of the segment.

${(x_m, y_m) = \left(\frac{0 + 9}{2}, \frac{-4 + 10}{2}\right)}$

Step 3: Simplify the Expression

We will simplify the expression to find the midpoint.

${(x_m, y_m) = \left(\frac{9}{2}, \frac{6}{2}\right)}$

${(x_m, y_m) = \left(\frac{9}{2}, 3\right)}$

Conclusion

In this article, we learned how to find the midpoint of a segment with given endpoints. We used the midpoint formula to find the midpoint of the segment with endpoints ${(0, -4)}$ and ${(9, 10)}$ . The midpoint of the segment is ${\left(\frac{9}{2}, 3\right)}$ . We hope this article has provided you with a clear understanding of how to find the midpoint of a segment with given endpoints.

Example Problems

Find the midpoint of a segment with endpoints ${(2, 5)}$ and ${(8, 12)}$ .
Find the midpoint of a segment with endpoints ${(-3, 7)}$ and ${(5, 9)}$ .

Solutions to Example Problems

Find the midpoint of a segment with endpoints ${(2, 5)}$ and ${(8, 12)}$ .

${(x_m, y_m) = \left(\frac{2 + 8}{2}, \frac{5 + 12}{2}\right)}$

${(x_m, y_m) = \left(\frac{10}{2}, \frac{17}{2}\right)}$

${(x_m, y_m) = \left(5, \frac{17}{2}\right)}$

Find the midpoint of a segment with endpoints ${(-3, 7)}$ and ${(5, 9)}$ .

${(x_m, y_m) = \left(\frac{-3 + 5}{2}, \frac{7 + 9}{2}\right)}$

${(x_m, y_m) = \left(\frac{2}{2}, \frac{16}{2}\right)}$

${(x_m, y_m) = \left(1, 8\right)}$

Applications of Midpoint Formula

The midpoint formula has numerous applications in mathematics and real-life situations. Some of the applications of the midpoint formula include:

Finding the midpoint of a line segment with given endpoints.
Finding the midpoint of a line segment with given coordinates.
Finding the midpoint of a line segment with given lengths and angles.
Finding the midpoint of a line segment with given slopes and y-intercepts.

Real-Life Applications of Midpoint Formula

The midpoint formula has numerous real-life applications. Some of the real-life applications of the midpoint formula include:

Finding the midpoint of a line segment with given coordinates to determine the location of a point on a map.
Finding the midpoint of a line segment with given lengths and angles to determine the length and angle of a line segment.
Finding the midpoint of a line segment with given slopes and y-intercepts to determine the slope and y-intercept of a line segment.

Conclusion

Introduction

In our previous article, we learned how to find the midpoint of a segment with given endpoints using the midpoint formula. In this article, we will answer some frequently asked questions about the midpoint formula and provide additional examples to help you understand the concept better.

Q&A

Q1: What is the midpoint formula?

A1: The midpoint formula is a mathematical formula used to find the midpoint of a line segment with given endpoints. It is given by:

${(x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$

where ${(x_1, y_1)}$ and ${(x_2, y_2)}$ are the given endpoints of the segment.

Q2: How do I find the midpoint of a line segment with given endpoints?

A2: To find the midpoint of a line segment with given endpoints, you can use the midpoint formula. Simply substitute the given endpoints into the formula and simplify the expression to find the midpoint.

Q3: What if the endpoints of the line segment are not given?

A3: If the endpoints of the line segment are not given, you cannot find the midpoint using the midpoint formula. However, you can use other methods such as finding the midpoint of a line segment with given coordinates or using the distance formula to find the midpoint.

Q4: Can I use the midpoint formula to find the midpoint of a line segment with given lengths and angles?

A4: Yes, you can use the midpoint formula to find the midpoint of a line segment with given lengths and angles. However, you will need to use trigonometry to find the coordinates of the endpoints of the line segment.

Q5: How do I find the midpoint of a line segment with given slopes and y-intercepts?

A5: To find the midpoint of a line segment with given slopes and y-intercepts, you can use the midpoint formula. However, you will need to find the coordinates of the endpoints of the line segment using the slope-intercept form of a line.

Q6: Can I use the midpoint formula to find the midpoint of a line segment with given coordinates?

A6: Yes, you can use the midpoint formula to find the midpoint of a line segment with given coordinates. Simply substitute the given coordinates into the formula and simplify the expression to find the midpoint.

Q7: What if the line segment is not a straight line?

A7: If the line segment is not a straight line, you cannot use the midpoint formula to find the midpoint. However, you can use other methods such as finding the midpoint of a line segment with given coordinates or using the distance formula to find the midpoint.

Q8: Can I use the midpoint formula to find the midpoint of a line segment with given lengths and angles in 3D space?

A8: Yes, you can use the midpoint formula to find the midpoint of a line segment with given lengths and angles in 3D space. However, you will need to use vector operations to find the coordinates of the endpoints of the line segment.

Additional Examples

Find the midpoint of a line segment with endpoints ${(2, 5)}$ and ${(8, 12)}$ .
Find the midpoint of a line segment with endpoints ${(-3, 7)}$ and ${(5, 9)}$ .
Find the midpoint of a line segment with endpoints ${(0, 0)}$ and ${(3, 4)}$ .

Solutions to Additional Examples

Find the midpoint of a line segment with endpoints ${(2, 5)}$ and ${(8, 12)}$ .

${(x_m, y_m) = \left(\frac{2 + 8}{2}, \frac{5 + 12}{2}\right)}$

${(x_m, y_m) = \left(\frac{10}{2}, \frac{17}{2}\right)}$

${(x_m, y_m) = \left(5, \frac{17}{2}\right)}$

Find the midpoint of a line segment with endpoints ${(-3, 7)}$ and ${(5, 9)}$ .

${(x_m, y_m) = \left(\frac{-3 + 5}{2}, \frac{7 + 9}{2}\right)}$

${(x_m, y_m) = \left(\frac{2}{2}, \frac{16}{2}\right)}$

${(x_m, y_m) = \left(1, 8\right)}$

Find the midpoint of a line segment with endpoints ${(0, 0)}$ and ${(3, 4)}$ .

${(x_m, y_m) = \left(\frac{0 + 3}{2}, \frac{0 + 4}{2}\right)}$

${(x_m, y_m) = \left(\frac{3}{2}, \frac{4}{2}\right)}$

${(x_m, y_m) = \left(\frac{3}{2}, 2\right)}$

Conclusion

In this article, we answered some frequently asked questions about the midpoint formula and provided additional examples to help you understand the concept better. We hope this article has provided you with a clear understanding of how to use the midpoint formula to find the midpoint of a line segment with given endpoints.