Problem 62 - Algebra and Trigonometry

Question

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

Point B is at coordinates (10, 13).

Step 1: Understanding Midpoint Formula

To find the coordinates of point B, we need to use the formula for the midpoint of a line segment. The midpoint formula is \( M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \), where \((x_1, y_1)\) are the coordinates of point A, and \((x_2, y_2)\) are the coordinates of point B.

Step 2: Set Up Midpoint Equations

Since \( M(6,8) \) is the midpoint of segment \( AB \), we set up the equations using the midpoint formula: \( \frac{2 + x_2}{2} = 6 \) and \( \frac{3 + y_2}{2} = 8 \). These equations allow us to solve for the coordinates of \( B(x_2, y_2) \).

Step 3: Solve for x-coordinate of B

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

Step 4: Solve for y-coordinate of B

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

Step 5: Find Coordinates of B

Combine the results from Steps 3 and 4: The coordinates of point B are \( (10, 13) \).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot points, lines, and curves. It's made up of two number lines that cross each other, usually at a right angle. Think of one line going left and right (the x-axis) and the other going up and down (the y-axis). These axes help us locate points by using a pair of numbers, which we call coordinates. For example, the point \( M(6,8) \) means 6 steps along the x-axis and 8 steps up along the y-axis.Using a coordinate plane is super helpful for visualizing geometric ideas. When you plot points like \( M(6,8) \) and \( A(2,3) \), you can actually see the distance and relationship between them. This visualization makes it easier to understand how to find the midpoint or other calculations needed. Coordinates work like a map, guiding you to the precise location of a point on the plane.

Midpoint Equation

The midpoint formula is a nifty tool in geometry. It helps you find the middle point of a line segment. Imagine you have two points, \((x_1, y_1)\) and \((x_2, y_2)\). The midpoint \( M(x, y) \) is found using this formula: \[M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Using this formula, you "average" the x-coordinates and the y-coordinates separately. This gives you the coordinates of the midpoint, which is exactly halfway in between the two points.In the exercise, we're told \( M(6,8) \) is the midpoint between \( A \) and an unknown \( B \). By plugging the known values into the formula, you set up equations. Solving these equations reveals the coordinates of \( B \). This method transforms a seemingly complicated problem into manageable algebraic steps.

Solving Linear Equations

Solving linear equations is like playing detective to find the value of an unknown number, called a variable. The most fun part? You can use simple operations like addition, subtraction, multiplication, and division.In our exercise, the problem of finding point \( B \) boils down to solving two simple linear equations:

• \( \frac{2 + x_2}{2} = 6 \)
• \( \frac{3 + y_2}{2} = 8 \)

Each equation corresponds to one coordinate of point \( B \). To solve these, you first multiply both sides by 2 to get rid of the fraction:

• For \( x \): \( 2 + x_2 = 12 \)
• For \( y \): \( 3 + y_2 = 16 \)

Next, you isolate the variable by doing the reverse operation of what's happening to it:

• Subtract 2 from both sides for \( x \) to solve for \( x_2: x_2 = 10 \)
• Subtract 3 from both sides for \( y \) to solve for \( y_2: y_2 = 13 \)

This systematic approach makes finding unknown coordinates streamlined and straightforward.