Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that the midpoints of the four sides of an arbitrary quadrilateral are the vertices of a parallelogram.

Step-by-Step Solution

Answer

The midpoints of a quadrilateral's sides form a parallelogram.

Step 1: Define the Quadrilateral

Consider a quadrilateral with vertices labeled as \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), and \( D(x_4, y_4) \). These are arbitrary points on a Cartesian coordinate plane.

Step 2: Calculate Midpoints

Find the midpoints of the sides of the quadrilateral. The midpoint \( M_1 \) of side \( AB \) is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \); midpoint \( M_2 \) of side \( BC \) is \( \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) \); midpoint \( M_3 \) of side \( CD \) is \( \left( \frac{x_3 + x_4}{2}, \frac{y_3 + y_4}{2} \right) \); and midpoint \( M_4 \) of side \( DA \) is \( \left( \frac{x_4 + x_1}{2}, \frac{y_4 + y_1}{2} \right) \).

Step 3: Vector Representation of Sides

Represent the lines between midpoints as vectors: vector \( \overrightarrow{M_1M_2} = \left( \frac{x_3 - x_1}{2}, \frac{y_3 - y_1}{2} \right) \), and \( \overrightarrow{M_3M_4} = \left( \frac{x_1 - x_3}{2}, \frac{y_1 - y_3}{2} \right) \).

Step 4: Calculate Opposing Sides

Check if the vectors \( \overrightarrow{M_1M_2} \) and \( \overrightarrow{M_3M_4} \) are equal in magnitude and direction. Simplifying, both are \( \left( \frac{x_3 - x_1}{2}, \frac{y_3 - y_1}{2} \right) \) but with opposite directions, thus equal and opposite.

Step 5: Check Another Pair of Sides

For the vectors \( \overrightarrow{M_2M_3} = \left( \frac{x_4 - x_2}{2}, \frac{y_4 - y_2}{2} \right) \) and \( \overrightarrow{M_4M_1} = \left( \frac{x_2 - x_4}{2}, \frac{y_2 - y_4}{2} \right) \). They are also equal in magnitude and opposite in direction.

Step 6: Conclusion

Since both pairs of opposite sides are equal in magnitude and opposite in direction, the quadrilateral formed by midpoints \( M_1, M_2, M_3, \) and \( M_4 \) is a parallelogram by definition.

Key Concepts

Parallelogram Properties

Parallelograms are specific quadrilaterals that boast several unique characteristics. Understanding these properties is essential in proving geometrical theorems, such as the one in the exercise. Here are the key features of a parallelogram:

• Opposite Sides Are Equal: In a parallelogram, opposite sides are not only parallel but also equal in length. This attribute is crucial for proving that the figure formed by the midpoints of a quadrilateral's sides is a parallelogram.


• Opposite Angles Are Equal: If a quadrilateral is a parallelogram, then its opposite angles are congruent.


• Diagonals Bisect Each Other: An important property is that the diagonals of a parallelogram bisect each other. This means that the intersection of the diagonals divides each diagonal into two equal parts.


• Consecutive Angles Are Supplementary: Any two consecutive angles in a parallelogram add up to 180 degrees. This is a direct result of the parallel sides with transversal lines.

These properties allow us to identify parallelograms in coordinate geometry and guide us in solving related problems effectively.

Quadrilateral Geometry

Quadrilaterals are polygons with four sides, and understanding their geometry is crucial to exploring their various types, including parallelograms, trapezoids, and rectangles. Here’s a simplified look at quadrilateral geometry:

• Sum of Angles: The total sum of the interior angles of any quadrilateral is always 360 degrees. This rule applies regardless of the shape or size of the quadrilateral.


• Types of Quadrilaterals: Quadrilaterals are categorized based on their sides and angles. Common types include rectangles, squares, trapezoids, and rhombuses, each having distinctive properties. Parallel sides, equal sides, or right angles help define these categories.


• Diagonals and Symmetry: The diagonals of a quadrilateral may or may not be equal, and their intersection points can provide insightful geometric properties.

Understanding the basics of quadrilateral geometry supports the proof of more complex theorems, like creating new figures from midpoint connections.

Coordinate Geometry Proofs

Coordinate geometry, also known as analytic geometry, uses algebraic techniques to solve geometric problems, making it a powerful tool in mathematics. Here's how coordinate geometry facilitates proofs:

• Using Coordinates: Any point in a plane can be represented as coordinates \(x, y\), allowing precise calculations of distances and slopes.


• Midpoint Formula: The midpoint \(M\) of a line segment between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is calculated as \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\), which is essential for constructing middles of quadrilateral sides.


• Vector Representation: Vectors help in clearly defining direction and magnitude, aiding in the proof of parallelism and equality, as seen with opposite sides of parallelograms.

Coordinate geometry proofs involve breaking down problems into algebraic expressions that can be solved, simplifying complex geometric constructions and proofs.