Which Of The Following Will Give The Location Of The Midpoint Of Segment AB, Given A ( 3 , 7 A (3,7 A ( 3 , 7 ] And B ( − 2 , 8 B (-2,8 B ( − 2 , 8 ]?A. \left(\frac{3+7}{2}, \frac{(-2)+8}{2}\right ]B. \left(\frac{8-7}{2}, \frac{3+2}{2}\right ]C.

Introduction

In geometry, a line segment is a part of a line that is bounded by two distinct points. The midpoint of a line segment is the point that divides the segment into two equal parts. In this article, we will explore how to find the midpoint of a line segment given the coordinates of its endpoints.

What is a Midpoint?

A midpoint is a point that lies exactly in the middle of a line segment. It is the point that divides the segment into two equal parts. The midpoint formula is used to find the coordinates of the midpoint of a line segment given the coordinates of its endpoints.

The Midpoint Formula

The midpoint formula is given by:

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

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

Example Problem

Let's consider the line segment AB with endpoints A(3, 7) and B(-2, 8). We want to find the midpoint of this line segment.

Step 1: Identify the Coordinates of the Endpoints

The coordinates of the endpoints of the line segment AB are:

• A(3, 7)
• B(-2, 8)

Step 2: Apply the Midpoint Formula

To find the midpoint of the line segment AB, we will apply the midpoint formula:

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

where $(x_1, y_1)$ = (3, 7) and $(x_2, y_2)$ = (-2, 8).

Substituting the values, we get:

$$\left(\frac{3+(-2)}{2}, \frac{7+8}{2}\right)$$

Simplifying the expression, we get:

$$\left(\frac{1}{2}, \frac{15}{2}\right)$$

Answer

The midpoint of the line segment AB is:

$$\left(\frac{1}{2}, \frac{15}{2}\right)$$

Conclusion

In this article, we have learned how to find the midpoint of a line segment given the coordinates of its endpoints. We have applied the midpoint formula to find the midpoint of the line segment AB with endpoints A(3, 7) and B(-2, 8). The midpoint of the line segment AB is $\left(\frac{1}{2}, \frac{15}{2}\right)$.

Common Mistakes to Avoid

When finding the midpoint of a line segment, it is essential to avoid common mistakes. Here are some common mistakes to avoid:

• Incorrect application of the midpoint formula: Make sure to apply the midpoint formula correctly by substituting the values of the coordinates of the endpoints.
• Incorrect simplification of the expression: Make sure to simplify the expression correctly by performing the arithmetic operations.
• Failure to check the units: Make sure to check the units of the coordinates of the endpoints and the midpoint.

Practice Problems

Here are some practice problems to help you practice finding the midpoint of a line segment:

• Find the midpoint of the line segment CD with endpoints C(4, 9) and D(1, 2).
• Find the midpoint of the line segment EF with endpoints E(-3, 5) and F(2, 1).
• Find the midpoint of the line segment GH with endpoints G(6, 3) and H(-1, 4).

Answer Key

Here are the answers to the practice problems:

• The midpoint of the line segment CD is $\left(\frac{5}{2}, \frac{11}{2}\right)$.
• The midpoint of the line segment EF is $\left(-\frac{1}{2}, 3\right)$.
• The midpoint of the line segment GH is $\left(\frac{5}{2}, \frac{7}{2}\right)$.

Final Thoughts

Introduction

In our previous article, we explored how to find the midpoint of a line segment given the coordinates of its endpoints. In this article, we will answer some frequently asked questions about the midpoint formula.

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 given the coordinates of its endpoints. The formula is:

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

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

Q: How do I apply the midpoint formula?

A: To apply the midpoint formula, you need to substitute the values of the coordinates of the endpoints into the formula. For example, if the endpoints of the line segment are A(3, 7) and B(-2, 8), you would substitute the values as follows:

$$\left(\frac{3+(-2)}{2}, \frac{7+8}{2}\right)$$

Simplifying the expression, you get:

$$\left(\frac{1}{2}, \frac{15}{2}\right)$$

Q: What if the coordinates of the endpoints are negative?

A: If the coordinates of the endpoints are negative, you can still apply the midpoint formula. For example, if the endpoints of the line segment are A(-3, -7) and B(2, -8), you would substitute the values as follows:

$$\left(\frac{-3+2}{2}, \frac{-7+(-8)}{2}\right)$$

Simplifying the expression, you get:

$$\left(-\frac{1}{2}, -\frac{15}{2}\right)$$

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

A: Yes, you can use the midpoint formula to find the midpoint of a line segment with coordinates in scientific notation. For example, if the endpoints of the line segment are A(3.2 x 10^2, 7.5 x 10^2) and B(-2.5 x 10^2, 8.2 x 10^2), you would substitute the values as follows:

$$\left(\frac{3.2 x 10^2 + (-2.5 x 10^2)}{2}, \frac{7.5 x 10^2 + 8.2 x 10^2}{2}\right)$$

Simplifying the expression, you get:

$$\left(0.35 x 10^2, 7.85 x 10^2\right)$$

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

A: Yes, you can use the midpoint formula to find the midpoint of a line segment with coordinates in decimal form. For example, if the endpoints of the line segment are A(3.2, 7.5) and B(-2.5, 8.2), you would substitute the values as follows:

$$\left(\frac{3.2 + (-2.5)}{2}, \frac{7.5 + 8.2}{2}\right)$$

Simplifying the expression, you get:

$$\left(0.35, 7.85\right)$$

Q: What if I have a line segment with coordinates in polar form? Can I use the midpoint formula?

A: Yes, you can use the midpoint formula to find the midpoint of a line segment with coordinates in polar form. However, you need to convert the polar coordinates to rectangular coordinates first. For example, if the endpoints of the line segment are A(3, 4) and B(-2, 8), you would convert the polar coordinates to rectangular coordinates as follows:

$$A(3, 4) \rightarrow A(3 \cos \theta, 3 \sin \theta)$$

$$B(-2, 8) \rightarrow B(-2 \cos \theta, -2 \sin \theta)$$

Then, you can apply the midpoint formula as follows:

$$\left(\frac{3 \cos \theta + (-2) \cos \theta}{2}, \frac{3 \sin \theta + (-2) \sin \theta}{2}\right)$$

Simplifying the expression, you get:

$$\left(\frac{1}{2} \cos \theta, \frac{1}{2} \sin \theta\right)$$

Conclusion

In this article, we have answered some frequently asked questions about the midpoint formula. We have explored how to apply the midpoint formula, how to handle negative coordinates, and how to use the midpoint formula with coordinates in scientific notation, decimal form, and polar form. By understanding the midpoint formula and its applications, you can find the midpoint of a line segment with ease.