Problem 56 - Algebra and Trigonometry

Question

Completing a Line Segment Plot the points \(M(6,8)\) and \(A(2,3)\) on a coordinate plane. If \(M\) is the midpoint of the line segment \(A B,\) find the coordinates of \(B\) . Write a brief description of the steps you took to find \(B,\) and your reasons for taking them.

Step-by-Step Solution

Answer

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

Step 1: Understand the Midpoint Formula

The midpoint formula is used to find the midpoint \((x_m, y_m)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\). The formula is \(x_m = \frac{x_1 + x_2}{2}\) and \(y_m = \frac{y_1 + y_2}{2}\).

Step 2: Substitute Known Values

Given that \(M(6,8)\) is the midpoint of the line segment \(A B\) and \(A(2,3)\), substitute the known values into the midpoint formula: \(6 = \frac{2 + x_2}{2}\) and \(8 = \frac{3 + y_2}{2}\).

Step 3: Solve for the x-coordinate of B

To find \(x_2\), solve the equation \(6 = \frac{2 + x_2}{2}\). Multiply both sides by 2 to get \(12 = 2 + x_2\), and then subtract 2 from both sides to find \(x_2 = 10\).

Step 4: Solve for the y-coordinate of B

To find \(y_2\), solve the equation \(8 = \frac{3 + y_2}{2}\). Multiply both sides by 2 to get \(16 = 3 + y_2\), and then subtract 3 from both sides to find \(y_2 = 13\).

Step 5: Write the Coordinates of B

After solving both equations, the coordinates of point \(B\) are determined to be \((10,13)\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent mathematical equations and points. It consists of two number lines: the horizontal axis is the x-axis, and the vertical axis is the y-axis. These axes intersect at a point known as the origin, which has the coordinates \((0,0)\).

Every point on this plane is represented by an ordered pair, \((x,y)\), where \(x\) indicates the horizontal position and \(y\) indicates the vertical position.

• The positive direction on the x-axis is to the right, while the negative direction is to the left.
• On the y-axis, positive values go up and negative values go down.

Understanding the coordinate plane allows us to easily plot and locate points, and analyze the relationships between different points.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints. It is the shortest path between those endpoints on the coordinate plane.

Unlike a line, which extends infinitely in both directions, a line segment has a defined length. In mathematical problems, line segments are often used to describe the distance or relationship between two points.

When working with line segments on a coordinate plane:

• The endpoints are identified using coordinates, such as \((x_1, y_1)\) and \((x_2, y_2)\).
• The properties of line segments, such as their length, can be calculated using formulas, like the distance formula.

Line segments are a fundamental concept in geometry and are essential for understanding more complex geometric shapes.

Midpoint

The midpoint of a line segment is the point that is exactly halfway between its endpoints. It divides the line segment into two equal parts. Calculating the midpoint is crucial in geometry, as it helps in various applications like bisecting angles and segments.

We use the midpoint formula to find this point's coordinates. Given endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((x_m, y_m)\) can be found using:\[x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}\]

In the exercise mentioned, point \(M(6,8)\) was given as the midpoint, which helped us derive the unknown endpoint \(B\). Knowing how to find a midpoint allows you to solve many geometric problems easily.

Plotting Points

Plotting points on a coordinate plane involves placing a dot at the location specified by an ordered pair, \((x, y)\). This activity is foundational for graphing equations and understanding geometric figures.

To plot a point:

• Start at the origin \((0,0)\), and move horizontally to the x-value.
• Then, move vertically to the y-value.
• Mark the intersection of these movements with a point.

In our problem, the points \(M(6,8)\) and \(A(2,3)\) are plotted first. This visualization helps in understanding their spatial relationship, allowing us to use the midpoint formula effectively. Being skilled at plotting points enhances your ability to interpret and solve geometric problems.