Problem 46 - Precalculus Mathematics for Calculus

Question

If \(M(6,8)\) is the midpoint of the line segment \(A B\) and if \(A\) has coordinates \((2,3),\) find the coordinates of \(B\)

Step-by-Step Solution

Answer

The coordinates of point B are \((10, 13)\).

Step 1: Understand the Formula

To find the midpoint of a line segment with endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\), use the formula for the midpoint: \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\). Here, the midpoint coordinates are \((6, 8)\), and point \(A\) is \((2, 3)\).

Step 2: Set Up Equations for the Midpoint

We know the midpoint coordinates \((6, 8)\). Set up the equation for the x-coordinate of the midpoint: \(\frac{2 + x_2}{2} = 6\). Set up the equation for the y-coordinate of the midpoint: \(\frac{3 + y_2}{2} = 8\).

Step 3: Solve for the x-coordinate of B

Solve the equation \(\frac{2 + x_2}{2} = 6\). Multiply both sides by 2: \(2 + x_2 = 12\). Subtract 2 from both sides: \(x_2 = 10\).

Step 4: Solve for the y-coordinate of B

Solve the equation \(\frac{3 + y_2}{2} = 8\). Multiply both sides by 2: \(3 + y_2 = 16\). Subtract 3 from both sides: \(y_2 = 13\).

Step 5: Write the Coordinates of B

Combine the solutions for \(x_2\) and \(y_2\) to write the coordinates of point \(B\). The coordinates of \(B\) are \((10, 13)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a fascinating branch of mathematics that provides a clear and precise way of representing geometric figures using coordinates. It allows us to study geometric figures through a coordinate system, enabling us to use algebra to solve geometric problems.

To put it simply, coordinate geometry involves using algebraic equations to describe the properties and relationships of geometric figures. This approach offers a significant advantage, as it unites topics of algebra and geometry, making it easier to visualize and solve problems using a dynamic method.

In the context of the exercise, coordinate geometry plays a vital role in finding the missing coordinates of a point when given the midpoint and one endpoint of a line segment. This is achieved by applying the midpoint formula within the coordinate plane.

Line Segment

A line segment is a fundamental part of geometry. It is a part of a line that is bounded by two distinct endpoints. Unlike a line that extends infinitely in both directions, a line segment has a fixed length determined by its endpoints.

In our exercise, the line segment is formed between points A and B with midpoint M. This is important when you need to determine the coordinates of any missing endpoint if the midpoint and another endpoint are known.

• The length of the line segment is calculated using the distance between its two endpoints.
• The midpoint divides the line segment into two equal parts.

Understanding line segments and their properties is crucial when applying the midpoint formula to find unknown coordinates, as you need to know both endpoints to comprehend the structure of the segment entirely.

Coordinate System

The coordinate system is an essential concept in coordinate geometry. It enables us to pinpoint locations on a plane using an ordered pair of numbers, typically known as coordinates. The most common type of coordinate system is the Cartesian coordinate system, named after the mathematician René Descartes.

In the Cartesian plane, each point is defined by an

• x-coordinate, which shows the point's horizontal position.
• y-coordinate, which shows the point's vertical position.

For instance, in the exercise, point A is located at

• (2, 3), reflecting its position on the plane as 2 units along the x-axis and 3 units along the y-axis.

The midpoint M (6, 8) divides the segment from A to B, helping us to determine the location of point B using the coordinate system's principles.

Problem Solving

Problem-solving is an indispensable skill, especially in mathematical exercises like those involving coordinate geometry. This skill allows you to apply theoretical concepts to practical situations.

When tasked with finding the coordinates of a point using the midpoint formula, approach the problem methodically:

• Identify the known values, like the coordinates of the midpoint and one endpoint.
• Apply the midpoint formula to set up equations based on the x and y coordinates.
• Solve these equations to find the missing coordinates.

For example, in this exercise, we begin by acknowledging that M is the midpoint, with known coordinates. By setting the midpoint formula equations, you can solve for the unknowns, carefully carrying through the arithmetic processes.

This logical process not only arrives at the correct solution but also deepens your understanding and competence in geometric problem-solving.