Question

The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. \((-1,-2),(-3,-6)\)

Short answer

The midpoint coordinates are (-2, -4).

Step-by-step solution

Understanding the Formula

The formula for finding the midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \( \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). This formula calculates the average of the x-coordinates and the y-coordinates.

Identify Endpoints

For this problem, the given endpoints are \((-1, -2)\) and \((-3, -6)\). Here, \(x_1 = -1\), \(y_1 = -2\), \(x_2 = -3\), and \(y_2 = -6\).

Calculate the Midpoint's x-coordinate

Apply the midpoint formula for the x-coordinate: \( \frac{x_1 + x_2}{2} = \frac{-1 + (-3)}{2} = \frac{-4}{2} = -2 \).

Calculate the Midpoint's y-coordinate

Apply the midpoint formula for the y-coordinate: \( \frac{y_1 + y_2}{2} = \frac{-2 + (-6)}{2} = \frac{-8}{2} = -4 \).

Combine Results

Combine the results of the x and y calculations to determine the midpoint coordinates. The midpoint of the segment is \((-2, -4)\).

Key concepts

Coordinate Geometry

Coordinate geometry, also called analytic geometry, is a fascinating branch of mathematics that allows us to use a coordinate system to explore and describe geometric figures. It combines algebra and geometry and enables precise measurements and calculations.

By placing figures on a grid of points determined by coordinates, it becomes easy to perform operations such as measuring distances between points, angles, and other properties. The coordinate plane is composed of two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical), forming a plane on which any point can be identified by a pair of numbers known as coordinates \((x, y)\).

This mathematical wonder is not just for the classroom but finds wide applications in computer graphics, robotics, geography, physics, and many other fields.

Finding Midpoints

Finding the midpoint is a basic yet essential concept in coordinate geometry. It's about locating a point that is exactly in the middle between two given points on the plane.

To find the midpoint, we use the midpoint formula. The midpoint formula helps us calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints. In mathematical terms, for any points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) is given by:
\[ M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \]
This formula tells us exactly where that balancing point, or midpoint, lies on the plane between the two original points.

Let's apply this idea to our problem where the endpoints are \( (-1, -2) \) and \( (-3, -6) \). Using the midpoint formula:

• The x-coordinate is \( \frac{-1 + (-3)}{2} = -2 \)
• The y-coordinate is \( \frac{-2 + (-6)}{2} = -4 \)

Thus, the midpoint coordinates are \( (-2, -4) \).

Geometry Lesson

In a typical geometry lesson focusing on coordinate geometry, you would immerse yourself in exploring shapes and lines using coordinates to understand better how they interact.

Understanding midpoints is a key part of these lessons. Midpoints help us in constructing bisectors, finding centers of figures like circles, and solving real-life problems that require balancing two points.

During exercises, always remember to:

• Identify the endpoints clearly before starting the calculation.
• Use the formula correctly by carefully substituting the values.
• Think of midpoints as the "average" location that balances your endpoints on a seesaw.

The lesson isn't only about calculating results but also about interpreting these results in real-world contexts. Through this learning process, you gain the skills to solve geometric problems with confidence and clarity.