Problem 18 - The Calculus with Analytic Geometry

Question

$$ \text { Three consecutive vertices of a parallelogram are }(-4,1),(2,3), \text { and }(8,9) \text {. Find the coordinates of the fourth vertex. } $$

Step-by-Step Solution

Answer

The coordinates of the fourth vertex are \((2, 7)\).

Step 1: Understand the problem

Three consecutive vertices of a parallelogram are given: \ \((-4,1)\), \((2,3)\), and \((8,9)\). Identify these as points \(A\), \(B\), and \(C\). The task is to find the fourth vertex \(D\).

Step 2: Use the properties of parallelograms

In a parallelogram, the diagonals bisect each other. This means the midpoint of \(AC\) is also the midpoint of \(BD\). Find the midpoint of \(AC\) first.

Step 3: Calculate the midpoint of \(AC\)

The coordinates of the midpoint of \(AC\) are found by averaging the x-coordinates and the y-coordinates of points \(A\) and \(C\): \[ \left( \frac{-4+8}{2}, \frac{1+9}{2} \right) = (2, 5). \]

Step 4: Set up the equations for midpoint of \(BD\)

Let \((x,y)\) be the coordinates of the fourth vertex \(D\). The midpoint of \(BD\) should also be \((2, 5)\): \[ \frac{2 + x}{2} = 2, \quad \frac{3 + y}{2} = 5. \]

Step 5: Solve for \(x\) and \(y\)

Solve the system of equations: \[ \frac{2 + x}{2} = 2 \ 2 + x = 4 \ x = 2. \ \frac{3 + y}{2} = 5 \ 3 + y = 10 \ y = 7. \]

Step 6: Write the coordinates of the fourth vertex

The coordinates of the fourth vertex \(D\) are \((2, 7)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes and their properties in the coordinate plane. By plotting points defined by their coordinates \((x, y)\), we can analyze various shapes such as lines, circles, and polygons.
For example, in this problem, we are given three vertices of a parallelogram in coordinate form. These coordinates help us employ mathematical tools, such as the midpoint formula, to solve for the unknown fourth vertex.
Understanding coordinate geometry is crucial because it provides a way to deal with geometric problems algebraically, allowing us to derive precise solutions through calculations.

Parallelogram Properties

A parallelogram is a quadrilateral with opposite sides that are equal in length and parallel. Several key properties are essential for solving problems involving parallelograms:

• Opposite sides are equal in length and parallel.
• Opposite angles are equal.
• Consecutive angles are supplementary (sum up to 180 degrees).
• Diagonals bisect each other at right angles.

In our problem, the key property we use is that diagonals bisect each other. This means that the midpoint of one diagonal is the same as the midpoint of the other diagonal. Using this property helps us find the coordinates of the fourth vertex in the parallelogram.

Midpoint Formula

The midpoint formula is used to find the center point between two given points in the coordinate plane. This formula is vital in coordinate geometry and is given by:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
In our example, we apply the midpoint formula to find the midpoint of diagonal AC, which is also the midpoint of diagonal BD. We calculate:
\[ \left( \frac{-4+8}{2}, \frac{1+9}{2} \right) = (2, 5) \]
Thus, the midpoint of AC is \((2, 5)\), and this same point will be the midpoint of BD, allowing us to set up equations to solve for the fourth vertex.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations simultaneously.
In the context of this problem, we set up a system of equations based on the midpoint we found:
\[ \frac{2 + x}{2} = 2 \]
\[ \frac{3 + y}{2} = 5 \]
Solving these equations step by step:
\[ 2 + x = 4 \quad \rightarrow x = 2 \]
\[ 3 + y = 10 \quad \rightarrow y = 7 \]
Thus, the coordinates of the fourth vertex D are \((2, 7)\). Understanding and solving systems of equations is fundamental for many problems in algebra and coordinate geometry.