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Magazine Chapter 2: Problem 19
Given \(A(-3,8),\) find the coordinates of the point \(B\) such that \(Q(5,-10)\) is the midpoint of segment \(A B\)
Short Answer
Expert verified
The coordinates of point B are (13, -28).
Step by step solution
01
Understanding Midpoint Formula
The coordinates of the midpoint of a line segment are calculated using the formula: \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Here, \(M\) is the midpoint, and \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints. In this problem, \(Q(5,-10)\) is the midpoint.
02
Identify Known Values
We are given that \(Q(5, -10)\) is the midpoint. Thus, \(x_M = 5\) and \(y_M = -10\). The coordinates of \(A\) are \((-3, 8)\), so \(x_1 = -3\) and \(y_1 = 8\). We need to find \((x_2, y_2)\), which represent the coordinates of point \(B\).
03
Set Up Equations for Midpoint Coordinates
Using the midpoint formula, set the equations: \(\frac{-3 + x_2}{2} = 5\) for the x-coordinate and \(\frac{8 + y_2}{2} = -10\) for the y-coordinate. These equations will help us find the unknown coordinates of \(B\).
04
Solve for x-coordinate
Solve the equation \(\frac{-3 + x_2}{2} = 5\) for \(x_2\). Multiply both sides by 2: \(-3 + x_2 = 10\). Add 3 to both sides: \(x_2 = 13\).
05
Solve for y-coordinate
Solve the equation \(\frac{8 + y_2}{2} = -10\) for \(y_2\). Multiply both sides by 2: \(8 + y_2 = -20\). Subtract 8 from both sides: \(y_2 = -28\).
06
Conclusion
The coordinates of point \(B\) are \((13, -28)\). These satisfy the given condition where \(Q(5, -10)\) is the midpoint of segment \(AB\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Geometry
Coordinate Geometry is a field of mathematics that helps us investigate the properties of geometric figures using a coordinate system.
It involves placing figures on a plane that is defined by a coordinate system, notably a Cartesian plane with x and y axes.
It involves placing figures on a plane that is defined by a coordinate system, notably a Cartesian plane with x and y axes.
- Coordinates are essential, as they show the precise position of points on this plane using pairs like (x, y).
- Coordinate Geometry enables us to calculate distances, midpoints, and slopes of lines by using algebraic methods.
- It reveals how rectangular grids allow true scientific exploration of lines, curves, and figures by translating geometric problems into algebraic equations.
- Understanding Coordinate Geometry is key for solving problems involving relationships between different points, such as finding a midpoint.
Line Segment
A Line Segment is a part of a line that is bounded by two distinct endpoints.
Unlike a line, which extends infinitely in both directions, a line segment has a fixed length.
Unlike a line, which extends infinitely in both directions, a line segment has a fixed length.
- Endpoints define the boundary, meaning all points on the segment lie between them.
- In our problem, segment AB is defined with points A and B.
- The Line Segment can be explored using its length, midpoint, or even its endpoint coordinates.
Midpoint Calculation
Midpoint Calculation is a process to find a point that is exactly in the middle of two other points on a line segment.
The Midpoint Formula is essential for this calculation:
The Midpoint Formula is essential for this calculation:
- Given two endpoints (x1, y1) and (x2, y2), the midpoint M is calculated as M = \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
- The Midpoint Formula helps in dividing a line segment into two equal parts, making it instrumental in various geometric constructions.
Point Coordinates
Point Coordinates refer to the ordered pair of numbers that define the exact position of a point on a coordinate plane.
Each point has a specific location based on two numbers: the x-coordinate (horizontal position) and the y-coordinate (vertical position).
Each point has a specific location based on two numbers: the x-coordinate (horizontal position) and the y-coordinate (vertical position).
- For example, point A has coordinates (-3, 8) which shows its precise location.
- Coordinates can represent both positive and negative positions depending on their locations on the Cartesian plane.
- In solving our exercise, understanding point B's coordinates help us to complete the definition of the line segment AB.
