Exercise 9.3:The term “midpoint” can have different meanings depending on the context. Here are the two most common ones:

**1. In geometry:**

The midpoint of a line segment is the **exact middle point** on that line segment. It’s the point that is **equidistant** (same distance) from both **endpoints** of the line segment.

Imagine a straight line with two points marking its ends. The midpoint would be located exactly in the **center** of that line, dividing it into two segments of **equal length**.

**2. In general:**

The midpoint can also refer to the **middle point** or **center point** of something in a more general sense. This could be the middle point of a time period, a set of data, a competition, or any other situation where there’s a starting and ending point and you want to identify the **central point**.Exercise 9.3:

The midpoint formula in coordinate geometry is a formula used to find the coordinates of the midpoint of a line segment when the coordinates of its endpoints are known. If the endpoints of a line segment are represented by the coordinates (x₁, y₁) and (x₂, y₂), then the coordinates of the midpoint (M) can be found using the following formula:

*M*(2*x*1+*x*2,2*y*1+*y*2)

So, the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.

#### Midpoint Formula in Coordinate Geometry

The midpoint formula in coordinate geometry is an equation used to find the coordinates of the **midpoint** of a line segment, given the coordinates of its **endpoints**.

Here’s the formula:

**Midpoint (x, y) = [(x1 + x2) / 2, (y1 + y2) / 2]**

where:

**(x1, y1)**and**(x2, y2)**are the coordinates of the two endpoints of the line segment.**(x, y)**represents the coordinates of the midpoint.

**Understanding the formula:**

**x1, x2**: These represent the x-coordinates of the two endpoints.**y1, y2**: These represent the y-coordinates of the two endpoints.**(x1 + x2) / 2**: This part calculates the average of the x-coordinates, which gives the x-coordinate of the midpoint.Exercise 9.3:**(y1 + y2) / 2**: This part calculates the average of the y-coordinates, which gives the y-coordinate of the midpoint.

**Examples:**

**Finding the midpoint of points (2, 5) and (6, 3):**

- Apply the formula:
- x = (2 + 6) / 2 = 4
- y = (5 + 3) / 2 = 4

- Therefore, the midpoint is
**(4, 4)**.

**Finding the midpoint of points (-3, 1) and (1, -5):**

- Apply the formula:
- x = (-3 + 1) / 2 = -1
- y = (1 – 5) / 2 = -2

- Therefore, the midpoint is
**(-1, -2)**.

#### Questions & Answers

**1. What is the midpoint formula in coordinate geometry?**The midpoint formula in coordinate geometry allows you to find the coordinates of the midpoint of a line segment when you know the coordinates of its two endpoints.

**2. How is the midpoint formula written?**The midpoint formula is:

(x, y) = ( (x1 + x2) / 2, (y1 + y2) / 2 )

where:

(x, y) represents the coordinates of the midpoint.

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

**3. How can you use the midpoint formula?**Substitute the known x and y coordinates of the two endpoints (x1, y1 and x2, y2) into the formula.

Perform the calculations to find the average values for both the x and y coordinates.

The resulting values, (x, y), represent the coordinates of the midpoint of the line segment.

**4. What is the geometric interpretation of the midpoint formula?**The midpoint formula essentially finds the average of the x and y coordinates of the two endpoints. This point lies exactly halfway between the two endpoints on the line segment.

**5. Can the midpoint formula be used for points in 3D space?**Yes, the concept of midpoint can be extended to 3D space. The formula for finding the midpoint of a line segment in 3D with endpoints (x1, y1, z1) and (x2, y2, z2) is:

(x, y, z) = ( (x1 + x2) / 2, (y1 + y2) / 2, (z1 + z2) / 2 )

**Q7: Are there alternative methods to find the midpoint without using the formula?**A7: Yes, another method involves averaging the corresponding coordinates separately. The x-coordinate of the midpoint is the average of the x-coordinates of the two points, and similarly for the y-coordinate. The formula provides a more concise way to achieve the same result.

#### Conclusion

The midpoint formula in coordinate geometry provides a **powerful and efficient** way to determine the **exact location** of the midpoint of a line segment, **solely based on the coordinates of its two endpoints**.

This formula serves as a valuable tool in various applications within **geometry** and other mathematical fields, allowing you to analyze and manipulate geometric shapes and relationships effectively.Exercise 9.3:

Must Read:

- Exercise.3.1: Scientific Notation
- Exercise.3.2: Common and Natural Logarithm
- Exercise.3.3: Laws of Logarithm
- Exercise.3.4: Application of Logarithm
- Exercise.5.1: Factorization
- Exploring Various Types of Factorization
- Exercise.5.3:Remainder Theorem And Factor Theorem
- Exercise.5.4: Factorization of a Cubic Polynomial
- Exercise.6.1: Highest Common Factor
- Method of Finding Least Common Multiple
- Exercise.6.2: Basic Operations on Algebraic Fractions
- Exercise.6.3: Square Root of Algebraic Expression

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