Question

The midpoint of the line segment from \(P_{1}\) to \(P_{2}\) is (-1,4) If \(P_{1}=(-3,6),\) what is \(P_{2} ?\)

Short answer

\(P_2 = (1, 2)\)

Step-by-step solution

Identify the midpoint formula

The midpoint formula for a line segment with endpoints \((x_1, y_1)\text{ and }(x_2, y_2)\) is given by: \[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Here, the midpoint \((M)\) is \((-1, 4)\) and one endpoint is \((x_1, y_1) = (-3, 6)\).

Set up the midpoint equations

Using the midpoint formula, set up two equations: \[\left( \frac{-3 + x_2}{2}, \frac{6 + y_2}{2} \right) = (-1, 4)\]This gives us two separate equations: 1. \frac{-3 + x_2}{2} = -12. \frac{6 + y_2}{2} = 4

Solve for \(x_2\)

Start with the first equation: \[\frac{-3 + x_2}{2} = -1\]Multiply both sides by 2: \[-3 + x_2 = -2\]Then add 3 to both sides: \[x_2 = 1\]

Solve for \(y_2\)

Now, solve the second equation: \[\frac{6 + y_2}{2} = 4\]Multiply both sides by 2: \[6 + y_2 = 8\]Subtract 6 from both sides: \[y_2 = 2\]

State the coordinates of \(P_2\)

The coordinates of \(P_2\) are found by combining the values of \( x_2 \) and \( y_2 \): \((1, 2)\).

Key concepts

Line Segments

In geometry, a line segment is a part of a line that has two endpoints. Imagine it like a piece of a road that starts at one point and ends at another. This is different from a line, which extends infinitely in both directions. Line segments are fundamental in understanding shapes and figures.

For example, if you have points \((P_1)\text{ and }(P_2)\) on a piece of paper, drawing a straight line from \((P_1)\text{ to }(P_2)\) forms a line segment. Unlike lines, line segments end where the endpoints are. These segments can be measured using rulers in basic geometry.

They are essential for calculating distances, understanding shapes, and even in constructing complex diagrams. In the context of algebra and coordinate geometry, line segments become useful for finding midpoints and other key geometric properties.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where points are defined and their properties investigated using a coordinate system. By placing points on a plane using an \((x, y)\) coordinate system, we can study geometrical relationships algebraically.

In this system, every point is given a unique address using an ordered pair of numbers. The first number (or coordinate) represents the horizontal position (x-axis), and the second number represents the vertical position (y-axis). This helps in precisely locating points, plotting shapes, and understanding spatial relationships.

One significant concept in coordinate geometry is the **midpoint**. The midpoint of a line segment is the point exactly halfway between the two endpoints. It’s calculated using the midpoint formula: \(\text{Midpoint} = \bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg)\text{ where }(x_1, y_1)\text{ and }(x_2, y_2)\)are the coordinates of the endpoints. This formula is especially useful in splitting segments equally and is frequently used in various geometric constructions.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two expressions. They are composed of variables, constants, and arithmetic operations. When solving problems in coordinate geometry, algebraic equations help in finding unknown coordinates or distances.

For instance, in the exercise given, we have to find the unknown coordinates \((x_2, y_2)\). Using the midpoint formula, we set up algebraic equations based on the given midpoint and one known endpoint. These equations allow us to solve for the unknown variables systematically:

\(\text{1.} \frac{-3 + x_2}{2} = -1\)
\(\text{2.} \frac{6 + y_2}{2} = 4\)

By isolating the variables and performing basic operations like multiplication and addition, we solve for \((x_2)\) and \((y_2)\). Algebra simplifies the steps needed to reach the solution and ensures accuracy. It transforms complex geometric problems into manageable algebraic tasks. Being proficient in using algebraic equations is crucial for mastering various concepts in mathematics.