Point \[$ T \$\], The Midpoint Of Segment \[$\overline{RS}\$\], Can Be Found Using The Formula:$\[ X = \left(\frac{1}{2}\right)(6-2) \\]What Are The Coordinates Of Point \[$ T \$\]?A. \[$ T(3,2) \$\] B. \[$

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, and understanding how to find the midpoint of a line segment is crucial in various mathematical applications. In this article, we will explore the formula for finding the midpoint of a line segment and use it to find the coordinates of point T, the midpoint of segment RS.

The Midpoint Formula

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

${ x = \left(\frac{1}{2}\right)(x_1 + x_2) }$

${ y = \left(\frac{1}{2}\right)(y_1 + y_2) }$

where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment.

Finding the Coordinates of Point T

We are given that point T is the midpoint of segment RS, and the coordinates of R and S are (2, 4) and (6, 2), respectively. To find the coordinates of point T, we can use the midpoint formula:

${ x = \left(\frac{1}{2}\right)(2 + 6) }$

${ x = \left(\frac{1}{2}\right)(8) }$

${ x = 4 }$

${ y = \left(\frac{1}{2}\right)(4 + 2) }$

${ y = \left(\frac{1}{2}\right)(6) }$

${ y = 3 }$

Therefore, the coordinates of point T are (4, 3).

Conclusion

In this article, we have explored the midpoint formula and used it to find the coordinates of point T, the midpoint of segment RS. The midpoint formula is a powerful tool in mathematics, and understanding how to use it is essential in various mathematical applications. We hope that this article has provided a clear and concise explanation of the midpoint formula and its application.

Example Problems

1. Find the midpoint of the line segment with endpoints (3, 5) and (7, 1).
2. Find the midpoint of the line segment with endpoints (2, 3) and (6, 6).
3. Find the midpoint of the line segment with endpoints (4, 2) and (8, 4).

Solutions

1. Using the midpoint formula, we get:

${ x = \left(\frac{1}{2}\right)(3 + 7) }$

${ x = \left(\frac{1}{2}\right)(10) }$

${ x = 5 }$

${ y = \left(\frac{1}{2}\right)(5 + 1) }$

${ y = \left(\frac{1}{2}\right)(6) }$

${ y = 3 }$

Therefore, the midpoint of the line segment is (5, 3).

2. Using the midpoint formula, we get:

${ x = \left(\frac{1}{2}\right)(2 + 6) }$

${ x = \left(\frac{1}{2}\right)(8) }$

${ x = 4 }$

${ y = \left(\frac{1}{2}\right)(3 + 6) }$

${ y = \left(\frac{1}{2}\right)(9) }$

${ y = 4.5 }$

Therefore, the midpoint of the line segment is (4, 4.5).

3. Using the midpoint formula, we get:

${ x = \left(\frac{1}{2}\right)(4 + 8) }$

${ x = \left(\frac{1}{2}\right)(12) }$

${ x = 6 }$

${ y = \left(\frac{1}{2}\right)(2 + 4) }$

${ y = \left(\frac{1}{2}\right)(6) }$

${ y = 3 }$

Therefore, the midpoint of the line segment is (6, 3).

Final Thoughts

Introduction

The midpoint formula is a fundamental concept in mathematics that allows us to find the coordinates of the midpoint of a line segment. In our previous article, we explored the midpoint formula and used it to find the coordinates of point T, the midpoint of segment RS. In this article, we will answer some frequently asked questions about the midpoint formula and provide additional examples to help you understand this concept better.

Q&A

Q: What is the midpoint formula?

A: The midpoint formula is a mathematical formula used to find the coordinates of the midpoint of a line segment. It is given by:

${ x = \left(\frac{1}{2}\right)(x_1 + x_2) }$

${ y = \left(\frac{1}{2}\right)(y_1 + y_2) }$

where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment.

Q: How do I use the midpoint formula?

A: To use the midpoint formula, you need to follow these steps:

1. Identify the coordinates of the two endpoints of the line segment.
2. Plug the coordinates into the midpoint formula.
3. Simplify the equation 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 integers, decimals, and fractions. You can plug in any type of coordinates into the formula and simplify the equation to find the coordinates of the midpoint.

Q: Can I use the midpoint formula to find the midpoint of a line segment with three or more endpoints?

A: No, the midpoint formula is only used to find the midpoint of a line segment with two endpoints. If you have a line segment with three or more endpoints, you need to use a different formula or method to find the midpoint.

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

A: The midpoint formula works with any type of coordinates, including negative coordinates. You can plug in negative coordinates into the formula and simplify the equation to find the coordinates of the midpoint.

Q: Can I use the midpoint formula to find the midpoint of a line segment in three dimensions?

A: No, the midpoint formula is only used to find the midpoint of a line segment in two dimensions. If you have a line segment in three dimensions, you need to use a different formula or method to find the midpoint.

Example Problems

1. Find the midpoint of the line segment with endpoints (3, 5) and (7, 1).
2. Find the midpoint of the line segment with endpoints (2, 3) and (6, 6).
3. Find the midpoint of the line segment with endpoints (4, 2) and (8, 4).

Solutions

1. Using the midpoint formula, we get:

${ x = \left(\frac{1}{2}\right)(3 + 7) }$

${ x = \left(\frac{1}{2}\right)(10) }$

${ x = 5 }$

${ y = \left(\frac{1}{2}\right)(5 + 1) }$

${ y = \left(\frac{1}{2}\right)(6) }$

${ y = 3 }$

Therefore, the midpoint of the line segment is (5, 3).

2. Using the midpoint formula, we get:

${ x = \left(\frac{1}{2}\right)(2 + 6) }$

${ x = \left(\frac{1}{2}\right)(8) }$

${ x = 4 }$

${ y = \left(\frac{1}{2}\right)(3 + 6) }$

${ y = \left(\frac{1}{2}\right)(9) }$

${ y = 4.5 }$

Therefore, the midpoint of the line segment is (4, 4.5).

3. Using the midpoint formula, we get:

${ x = \left(\frac{1}{2}\right)(4 + 8) }$

${ x = \left(\frac{1}{2}\right)(12) }$

${ x = 6 }$

${ y = \left(\frac{1}{2}\right)(2 + 4) }$

${ y = \left(\frac{1}{2}\right)(6) }$

${ y = 3 }$

Therefore, the midpoint of the line segment is (6, 3).

Final Thoughts

In conclusion, the midpoint formula is a powerful tool in mathematics that allows us to find the coordinates of the midpoint of a line segment. By understanding how to use the midpoint formula, we can solve a variety of mathematical problems and applications. We hope that this article has provided a clear and concise explanation of the midpoint formula and its application.