Question

The midpoint of the line segment from \(P_{1}\) to \(P_{2}\) is (5,-4) . If \(P_{2}=(7,-2),\) what is \(P_{1} ?\)

Short answer

The coordinates of \(P_1\) are \( (3, -6) \).

Step-by-step solution

Identify the midpoint formula

The midpoint of a line segment with endpoints \(P_{1} (x_1, y_1)\) and \(P_{2} (x_2, y_2)\) is calculated using the formula: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

Set up equations for the x-coordinate

Given the midpoint (5, -4) and endpoint \(P_{2} = (7, -2)\), set up the equation for the x-coordinate: \[ 5 = \frac{x_1 + 7}{2} \].

Solve for x-coordinate of \(P_1\)

Multiply both sides of the equation by 2 to get rid of the fraction: \[ 2 \times 5 = x_1 + 7 \] Thus, \(10 = x_1 + 7\).Subtract 7 from both sides to find \(x_1\): \[ x_1 = 10 - 7 \] Therefore, \(x_1 = 3\).

Set up equations for the y-coordinate

Using the same logic, we set up the equation for the y-coordinate: \[ -4 = \frac{y_1 - 2}{2} \].

Solve for y-coordinate of \(P_1\)

Multiply both sides by 2: \[ 2 \times (-4) = y_1 - 2 \] Thus, \( -8 = y_1 - 2\).Add 2 to both sides to find \(y_1\): \[ y_1 = -8 + 2 \] Therefore, \(y_1 = -6\).

State the coordinates of \(P_1\)

Using the solved x and y coordinates, we find that \(P_1 = (3, -6)\).

Key concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to investigate the geometric properties and relationships of shapes and points. The most common coordinate system is the Cartesian plane, where each point is defined by an ordered pair \( x,y \) of numbers. Coordinate geometry allows us to describe geometric shapes algebraically and to perform calculations such as finding distances between points, slopes of lines, and midpoints of segments. By converting geometric problems into algebraic ones, coordinate geometry bridges the gap between algebra and geometry and provides powerful tools for solving real-world problems.

solving equations

Solving algebraic equations is a fundamental skill in mathematics and involves finding the values of variables that satisfy the given mathematical statements. In the context of finding a midpoint, you'll often encounter linear equations that need to be solved.

• Start by isolating the variable by performing inverse operations on both sides of the equation.
• If fractions are involved, multiply through by the denominator to clear the fraction.
• Simplify the equation step by step, using addition, subtraction, multiplication, or division to isolate the variable.

Mastery of solving equations is essential for solving more complex problems in coordinate geometry.

midpoint calculation

Calculating the midpoint of a line segment is a common problem in coordinate geometry. The midpoint formula is derived from averaging the x-coordinates and the y-coordinates of the endpoints. Given two points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \), the midpoint \( M \) is calculated using:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].
In our exercise, knowing the midpoint allowed us to set up equations to find unknown coordinates. By following the formula, you ensure that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints and similarly for the y-coordinate. This makes midpoint calculation very straightforward and easy to perform.

algebraic manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions and equations to solve for unknown variables. This process often involves:

• Adding or subtracting terms on both sides of the equation to isolate variables.
• Multiplying or dividing both sides of the equation by the same number to simplify.
• Using properties of equality to maintain the balance of the equation.

In our exercise, we used algebraic manipulation to solve for unknown coordinates by systematically transforming the equations into simpler forms. This methodical approach ensures clarity and accuracy in finding the correct solution.