Select The Correct Answer.The Endpoints Of G H ‾ \overline{GH} G H Are G ( 14 , 3 G(14, 3 G ( 14 , 3 ] And H ( 10 , − 6 H(10, -6 H ( 10 , − 6 ]. What Is The Midpoint Of G H ‾ \overline{GH} G H ?A. ( 6 , − 15 (6, -15 ( 6 , − 15 ] B. \left(-2, -\frac{9}{2}\right ] C.

Introduction

In geometry, the midpoint of a line segment is the point that divides the segment into two equal parts. It is an essential concept in mathematics, particularly in coordinate geometry. In this article, we will explore the midpoint formula and its application to find the midpoint of a line segment given its endpoints.

What is the Midpoint Formula?

The midpoint formula is a mathematical formula used to find the midpoint of a line segment given its endpoints. The formula is as follows:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints of the line segment.

How to Find the Midpoint of a Line Segment

To find the midpoint of a line segment, we need to follow these steps:

1. Identify the coordinates of the endpoints of the line segment.
2. Plug the coordinates into the midpoint formula.
3. Simplify the expression to find the coordinates of the midpoint.

Example: Finding the Midpoint of $\overline{GH}$

Let's consider an example to illustrate the application of the midpoint formula. Suppose we have a line segment $\overline{GH}$ with endpoints $G(14, 3)$ and $H(10, -6)$. We want to find the midpoint of $\overline{GH}$.

Step 1: Identify the Coordinates of the Endpoints

The coordinates of the endpoints of $\overline{GH}$ are:

• $G(14, 3)$
• $H(10, -6)$

Step 2: Plug the Coordinates into the Midpoint Formula

Now, we plug the coordinates into the midpoint formula:

$$\left(\frac{14 + 10}{2}, \frac{3 + (-6)}{2}\right)$$

Step 3: Simplify the Expression

Simplifying the expression, we get:

$$\left(\frac{24}{2}, \frac{-3}{2}\right)$$

$$\left(12, -\frac{3}{2}\right)$$

Therefore, the midpoint of $\overline{GH}$ is $(12, -\frac{3}{2})$.

Conclusion

In conclusion, the midpoint formula is a powerful tool used to find the midpoint of a line segment given its endpoints. By following the steps outlined in this article, we can easily find the midpoint of a line segment. The midpoint formula is a fundamental concept in mathematics, and its application is essential in various fields, including geometry, trigonometry, and calculus.

Answer

The correct answer is:

• B. $\left(-2, -\frac{9}{2}\right)$

$$

Q&A: Midpoint Formula

Q: What is the midpoint formula?

A: The midpoint formula is a mathematical formula used to find the midpoint of a line segment given its endpoints. The formula is as follows:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints of the line segment.

Q: How do I find the midpoint of a line segment?

A: To find the midpoint of a line segment, you need to follow these steps:

1. Identify the coordinates of the endpoints of the line segment.
2. Plug the coordinates into the midpoint formula.
3. Simplify the expression to find the coordinates of the midpoint.

Q: What if the coordinates of the endpoints are not integers?

A: The midpoint formula works with any type of coordinates, including fractions and decimals. You simply need to plug the coordinates into the formula and simplify the expression.

Q: Can I use the midpoint formula to find the midpoint of a line segment in 3D space?

A: Yes, the midpoint formula can be extended to find the midpoint of a line segment in 3D space. The formula is as follows:

$$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$$

where $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of the endpoints of the line segment.

Q: What is the significance of the midpoint formula in real-world applications?

A: The midpoint formula has numerous applications in real-world scenarios, including:

• Geometry and trigonometry: The midpoint formula is used to find the midpoint of a line segment, which is essential in geometry and trigonometry.
• Computer graphics: The midpoint formula is used to create smooth curves and surfaces in computer graphics.
• Engineering: The midpoint formula is used to find the midpoint of a line segment, which is essential in engineering applications such as designing bridges and buildings.

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

A: Yes, the midpoint formula works with negative coordinates. You simply need to plug the coordinates into the formula and simplify the expression.

Q: What if I have a line segment with coordinates that are not in the same order?

A: The midpoint formula works regardless of the order of the coordinates. You simply need to plug the coordinates into the formula and simplify the expression.

Conclusion

In conclusion, the midpoint formula is a powerful tool used to find the midpoint of a line segment given its endpoints. By following the steps outlined in this article, you can easily find the midpoint of a line segment. The midpoint formula has numerous applications in real-world scenarios, including geometry, trigonometry, computer graphics, and engineering.

Frequently Asked Questions

• What is the midpoint formula?
  • The midpoint formula is a mathematical formula used to find the midpoint of a line segment given its endpoints.
• How do I find the midpoint of a line segment?
  • To find the midpoint of a line segment, you need to follow these steps: 1. Identify the coordinates of the endpoints of the line segment. 2. Plug the coordinates into the midpoint formula. 3. Simplify the expression to find the coordinates of the midpoint.
• Can I use the midpoint formula to find the midpoint of a line segment in 3D space?
  • Yes, the midpoint formula can be extended to find the midpoint of a line segment in 3D space.

Additional Resources

• Midpoint Formula Calculator: A calculator that can be used to find the midpoint of a line segment given its endpoints.
• Midpoint Formula Worksheet: A worksheet that provides practice problems for finding the midpoint of a line segment using the midpoint formula.
• Midpoint Formula Video Tutorial: A video tutorial that provides a step-by-step guide on how to find the midpoint of a line segment using the midpoint formula.