Exercise 9.3: Mid-Point Formula Coordinate Geometry

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(2x1+x2,2y1+y2)

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).

Exercise 9.3: Mid-Point Formula Coordinate Geometry

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:

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