Question

Algebra Suppose \(C(-4,5)\) is the midpoint of \(\overline{A B}\) and the coordinates of \(A\) are \((2,17)\). Find the coordinates of \(B\).

Short answer

The coordinates of point B are \((-10, -7)\).

Step-by-step solution

Understanding the Midpoint Formula

The midpoint formula is used to find a point that is exactly halfway between two points in a Cartesian plane. The formula is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points A and B.

Applying the Midpoint Formula

Given that \( C(-4, 5) \) is the midpoint of \( \overline{AB} \), we can use the midpoint formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (-4, 5) \), where \((x_1, y_1) = (2, 17)\) and \((x_2, y_2)\) are the coordinates of point B.

Setting Up the Equations

Use the midpoint formula for each coordinate:1. For the x-coordinate:\[ \frac{2 + x_2}{2} = -4 \]2. For the y-coordinate:\[ \frac{17 + y_2}{2} = 5 \]

Solving for the x-coordinate

Solve the equation \( \frac{2 + x_2}{2} = -4 \) for \( x_2 \).Multiply both sides by 2 to clear the fraction:\[ 2 + x_2 = -8 \]Subtract 2 from both sides:\[ x_2 = -10 \]

Solving for the y-coordinate

Solve the equation \( \frac{17 + y_2}{2} = 5 \) for \( y_2 \).Multiply both sides by 2 to clear the fraction:\[ 17 + y_2 = 10 \]Subtract 17 from both sides:\[ y_2 = -7 \]

Finding the Coordinates of Point B

The coordinates of point B are the solutions from the previous steps. Thus, we have point B as \( (-10, -7) \).

Key concepts

Coordinate Geometry

Coordinate geometry is an area of mathematics that uses the principles of geometry alongside algebra. It's all about describing geometric figures and understanding their properties through numbers. One key element in coordinate geometry is the Cartesian plane. This plane is where you plot and connect points using ordered pairs. In coordinate geometry, the location of any point is identified by its x-coordinate (horizontal position) and y-coordinate (vertical position).

Using the Cartesian plane allows us to find distances, midpoints, and slopes of lines. By transforming geometric shapes into algebraic equations, we can handle complex calculations with relative ease. An essential tool in this field is the midpoint formula, which helps find the exact middle point between two given points. This formula is practical in solving problems like the one in our example, where we're tasked to locate the missing point B, knowing that point C is the midpoint of line segment \(\overline{AB}\).

Additionally, coordinate geometry helps in visualizing geometric problems. By representing data visually, one can see relationships and patterns that may not be evident through numbers alone. Understanding this concept is vital for solving equations that arise from real-world problems. It's a bridge between practical applications and theoretical understanding.

Solving Equations

Solving equations is a fundamental concept in algebra and a significant part of coordinate geometry. In the exercise, we apply the midpoint formula, which leads to equations for the coordinates of point B. Solving these equations step by step is crucial. Let's break it down:

• The key is to isolate the variable you're solving for, ensuring you simplify the equation progressively.
• For each equation derived from the midpoint formula, namely \( \frac{2 + x_2}{2} = -4 \) and \( \frac{17 + y_2}{2} = 5 \), the first step is to clear the fractions by multiplying both sides by 2.
• This leaves us with simpler equations: \( 2 + x_2 = -8 \) and \( 17 + y_2 = 10 \).
• Subtracting the constant from both sides solves for \(x_2\) and \(y_2\), yielding \(-10\) and \(-7\) respectively.

Understanding each operation helps to grasp how changes to one side of the equation affect the other. It's about maintaining balance on both sides while arriving at the solution. Solving these equations assists in finding the unknown coordinates of point B.

This skill is foundational in mathematics and extends to various fields like physics, engineering, and computer science. Mastery of solving equations widens your ability to tackle a wide range of mathematical challenges.

Cartesian Plane

The Cartesian plane is the stage where coordinate geometry unfolds. It revolutionized mathematics by allowing algebraic equations to be visualized graphically. This plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The intersection of these axes is known as the origin, denoted by the point \((0, 0)\).

Each point on this plane is represented by an ordered pair \((x, y)\). For instance, point \(A\) is represented as \((2, 17)\), and point \(C\) as \((-4, 5)\). When plotting or locating these points, you move horizontally along the x-axis and vertically along the y-axis. This system makes dealing with geometric problems more manageable, as complex shapes and lines convert into visual graphs.

The Cartesian plane serves as a basis for numerous mathematical concepts and real-world applications. It's essential for graphing linear equations, which appear as straight lines. Moreover, it helps in calculating distances using the distance formula, understanding symmetry, and much more.

Overall, the Cartesian plane's ability to tie together algebra and geometry makes it an incredibly powerful tool in mathematics. It not only aids in understanding theoretical concepts but also in practical problem-solving.