Question

The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. \((-18,5),(-3,-16)\)

Short answer

The midpoint of the segment is (-10.5, -5.5).

Step-by-step solution

Review Endpoint Coordinates

The endpoints of the segment are - First endpoint coordinates: \((-18, 5)\) - Second endpoint coordinates: \((-3, -16)\).We will use these two points to find the midpoint.

Apply Midpoint Formula

The midpoint \((M)\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:\[(x_1, y_1) = (-18, 5),\quad (x_2, y_2) = (-3, -16)\]\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

Substitute Endpoint Coordinates into Formula

Substitute the endpoint coordinates into the midpoint formula:\[M = \left( \frac{-18 + (-3)}{2}, \frac{5 + (-16)}{2} \right)\]

Perform the Calculations

Calculate each coordinate of the midpoint separately:- For the x-coordinate: \[\frac{-18 + (-3)}{2} = \frac{-21}{2} = -10.5\]- For the y-coordinate: \[\frac{5 + (-16)}{2} = \frac{-11}{2} = -5.5\]So, the midpoint \(M\) is \((-10.5, -5.5)\).

Key concepts

Coordinates

In coordinate geometry, coordinates are a set of values that show an exact position on the coordinate plane. A typical coordinate pair is written as \(x, y\), where \(x\) represents the horizontal position and \(y\) represents the vertical position. This two-dimensional system allows us to pinpoint precise locations on the plane, such as the coordinates of endpoints.

For example, consider the coordinates given in our exercise, \((-18, 5)\) and \((-3, -16)\). These values tell us exactly where these points are in relation to the horizontal x-axis and vertical y-axis. Understanding coordinates is essential for solving problems in geometry as it forms the basis for finding relationships between different points on the plane.

Endpoints

Endpoints are the two extreme points that define a line segment or any section of a geometric figure. In the context of our problem, we identify two endpoints: \((-18, 5)\) and \((-3, -16)\). These are crucial in determining the characteristics of the segment, including its midpoint.

To find the midpoint, you'll need the coordinates of these endpoints. Think of them as the start and end of the segment, and your task is to determine the exact middle between these two points. Knowing the endpoints helps in various calculations like distance, midpoint, or even finding the direction or slope of the line segment.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric shapes and transformations on a coordinate plane. It merges geometric concepts with algebraic techniques, making it a powerful tool for solving complex geometric problems.

By applying formulas like the midpoint formula \((M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}))\), we can find the midpoint of a segment formed by two endpoints. This method of using algebraic formulas is what makes coordinate geometry versatile and applicable in numerous mathematical tasks.

• Coordinate geometry simplifies the handling of geometric figures by converting them into algebraic expressions.
• It allows for precise calculations of distances, midpoints, and slopes, which are critical in higher-level mathematics and practical applications such as engineering and computer graphics.
• Learning coordinate geometry enhances spatial understanding by connecting abstract algebraic methods with tangible geometric visualization.