Question

Find the midpoint of the line segment joining the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(-2,0) ; \quad P_{2}=(2,4) $$

Short answer

The midpoint is \(M = (0, 2)\).

Step-by-step solution

Identify the coordinates of the points

The coordinates of the points are given as follows:Point \(P_{1} = (-2, 0)\)Point \(P_{2} = (2, 4)\)

Apply the midpoint formula

The formula for the midpoint \(M\) of a line segment joining the points \(P_{1}(x_{1}, y_{1})\) and \(P_{2}(x_{2}, y_{2})\) is given by:\[M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right)\]

Substitute the coordinates

Substitute the coordinates of \(P_{1}\) and \(P_{2}\) into the midpoint formula:\[M = \left( \frac{-2 + 2}{2}, \frac{0 + 4}{2} \right)\]

Simplify the expression

Simplify the expression to find the coordinates of the midpoint:\[M = \left( \frac{0}{2}, \frac{4}{2} \right) = (0, 2)\]

Key concepts

Line Segment

A line segment is a part of a line that is bounded by two distinct end points. Imagine a piece of spaghetti. You can break it into segments. These segments have a start and an end point. A line segment, in math, works the same way. It's a straight path with two end points. Understanding this helps us figure out various properties like lengths and midpoints. When we talk about midpoints, we mean the exact middle of this segment.

Coordinates

Coordinates are a set of values that show a specific position in a space. They are written as pairs like \((x, y)\). Each pair tells us where a point is on a grid. Think of it like a treasure map where the coordinates direct you to a precise location. In our example, \( P_1(-2, 0) \) and \( P_2(2, 4) \) are coordinates. The first number \(x\) tells us the horizontal position, while the second number \(y\) tells us the vertical position.

Algebra

Algebra deals with symbols and the rules for manipulating those symbols. In the context of the midpoint formula, algebra helps us to manipulate and solve equations more easily. For example, to find the midpoint of a line segment between two points, we use the formula: \ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right)\. This formula comes from adding the \(x\) coordinates of both points, dividing by 2, and doing the same for the \(y\) coordinates. It’s a simple but powerful application of algebra.