Question

Explain why it is correct to say that the coordinates of the midpoint of a segment are the means of the coordinates of the endpoints of the segment.

Short answer

The midpoint is at the coordinates' means because it equally divides the segment.

Step-by-step solution

Understand the Midpoint Concept

The midpoint of a segment is a point that divides the segment into two equal parts. If you have a segment with endpoints at coordinates \((x_1, y_1)\) and \((x_2, y_2)\), then the midpoint is a new point which is equidistant from both these endpoints.

Recall the Mean Definition

The arithmetic mean of two numbers is the sum of the numbers divided by two. For example, the mean of two numbers \(a\) and \(b\) is \(\frac{a+b}{2}\). This definition helps us find the average or central value between two numbers.

Calculate the Midpoint Coordinates

Using what we understand about midpoints and means, the midpoint of the segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated by taking the mean of the \(x\)-coordinates and the \(y\)-coordinates. The formula for the midpoint \(M(x, y)\) is given by: \[ M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]. This formula represents the means of the endpoint coordinates.

Explain Midpoint Formula

The midpoint formula works because it uses the arithmetic mean to find the halfway values for each coordinate. For the \(x\)-coordinates, it takes the average of \(x_1\) and \(x_2\), which gives the \(x\)-coordinate of the midpoint. Similarly, it takes the average of \(y_1\) and \(y_2\) for the \(y\)-coordinate. These averages ensure the point is equidistant from both endpoints, making it the precise midpoint of the segment.

Key concepts

Arithmetic Mean

The concept of arithmetic mean is quite essential, especially when dealing with points and segments in geometry. Simply put, the arithmetic mean of two numbers is what you commonly know as the average. To calculate it, you add the two values and then divide the result by two. For geometry applications, such as finding midpoint coordinates, this mean gives you a central or middle point between the two numbers.

• For example, the arithmetic mean of numbers 3 and 7 is calculated as \( rac{3 + 7}{2} = 5 \), making 5 the middle value.
• In coordinate geometry, the arithmetic mean helps find a point exactly in the middle of two endpoints, ensuring that the midpoint is evenly spaced between them along the segment.

Understanding the role of arithmetic mean is crucial for grasping how the midpoint formula functions. By averaging the coordinates, we ensure precision in locating the midpoint.

Coordinates

Coordinates express the position of points on a plane using a pair of numbers, usually noted as \( (x, y) \). The first number in the pair (\((x)\)) represents the horizontal position (along the x-axis), while the second number \( (y) \) represents the vertical position (along the y-axis).

• Coordinates allow you to pinpoint any location on a two-dimensional plane.
• They are crucial for creating shapes, geometrical figures, and defining the position of a segment's endpoints.
• For example, a point with coordinates \( (4, -3) \) is located 4 units right and 3 units down from the origin.

To find a midpoint, knowing the coordinates of the endpoints is essential. It allows you to calculate exactly where the segment is divided in half by adjusting each dimension independently.

Segment Endpoint

Segment endpoints are the points at both ends of a line segment. Each endpoint can be defined in the coordinate plane by its coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \). The segment itself consists of all points that lie on the straight path between these endpoints.

• The endpoints are the extreme points establishing the limits of a segment.
• In essence, they anchor the segment, and the line between them represents the shortest path connecting these points.

When focusing on midpoints, the coordinates of the endpoints play a significant role. By using these coordinates, not only can you visualize the segment, but you can apply the midpoint formula to determine precisely where the middle is.
The endpoint coordinates provide the necessary data to compute symmetrical or average points between the two extremes using arithmetic means. This concept helps you determine the midpoint effectively.