Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Midpoints form a parallelogram by equal opposite sides.

Step 1: Define Midpoints

Given a quadrilateral ABCD, let's identify the midpoints of its sides. Let M, N, P, and Q be the midpoints of sides AB, BC, CD, and DA, respectively.

Step 2: Apply Midpoint Formula

Apply the midpoint formula to find the coordinates of the midpoints. For example, for a line segment AB with endpoints A(x1, y1) and B(x2, y2), the midpoint M is given by \( M = \left( \frac{x1+x2}{2}, \frac{y1+y2}{2} \right) \). Do this for all midpoints (M, N, P, Q).

Step 3: Relate Midpoints to Vector Addition

Consider vectors \( \vec{MN} \) and \( \vec{PQ} \). By finding the vectors from the midpoint coordinates, \( \vec{MN} = N - M \) and \( \vec{PQ} = Q - P \). Note that \( \vec{MN} = \vec{PQ} \) due to the properties of vector addition and the fact that both represent one of the diagonals of the rectangle formed by connecting midpoints.

Step 4: Repeat for Diagonal Comparison

Similarly, analyze and compare vectors \( \vec{MP} \) and \( \vec{NQ} \) using the relationship from their coordinates: \( MP \) and \( NQ \). Show that \( \vec{MP} = \vec{NQ} \).

Step 5: Conclude Parallelogram Properties

Since both pairs of opposite sides are equal \( \vec{MN} = \vec{PQ} \) and \( \vec{MP} = \vec{NQ} \), conclude that \( MNPQ \) forms a parallelogram, as opposite sides are equal in length and parallel.

Key Concepts

Understanding Quadrilateral Midpoints

The concept of quadrilateral midpoints is fundamental when investigating the properties and relationships within a quadrilateral. When you connect the midpoints of all four sides of any quadrilateral, intriguing things happen. Imagine you have a quadrilateral, which is a four-sided figure, with vertices labeled A, B, C, and D. The midpoints of the sides, say M, N, P, and Q, divide each side into two equal parts. By understanding these midpoints, we delve into wonderful geometric properties that can transform complex shapes into simpler ones, like parallelograms.

• Midpoint M divides side AB into two equal segments.
• Similarly, N, P, and Q divide their respective sides BC, CD, and DA.

These midpoints are keys to unlocking interesting geometric proofs and spatial understanding.

Applying the Midpoint Formula

The midpoint formula is a mathematical tool used to find the exact middle point of a line segment. Given two endpoints, this formula allows you to calculate the coordinates of the midway point easily. For a line segment with endpoints A(x_1, y_1) and B(x_2, y_2), the midpoint M can be determined using the formula:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This concept is crucial in identifying the midpoints of the quadrilateral sides. For example:

• M is the midpoint of side AB.
• N is the midpoint of side BC.

Using this approach, you transform vertex connections into simpler, manageable components, paving the way for more complex geometric analyses.

Exploring Vector Addition

Vector addition is a powerful method for analyzing geometric properties, such as parallel lines and relative positioning. Vectors express quantities with both direction and magnitude, and they play a crucial role in proving relationships in geometry. In the context of quadrilateral midpoints forming a parallelogram, consider the vectors \( \vec{MN} \) and \( \vec{PQ} \). By using coordinates of these vectors:

• Calculate \( \vec{MN} = N - M \)
• Calculate \( \vec{PQ} = Q - P \)

In this context, the equality \( \vec{MN} = \vec{PQ} \) highlights that the line segments are equal in length and parallel, supporting the property that these segments form part of a parallelogram.

Thus, vector addition helps bridge midpoints into spatial relationships confirming geometric configurations.

A Stroll Through Parallelogram Proof

Proving that certain geometric shapes hold specific properties is a fascinating part of geometry. In this instance, if you connect the midpoints of all four sides of a quadrilateral, you form a parallelogram. Here’s how you establish and prove this:

Start by demonstrating that opposite vectors, like \( \vec{MN} \) and \( \vec{PQ} \), are both parallel and of equal length. Then, check the other pair of opposite sides, \( \vec{MP} \) and \( \vec{NQ} \). Once you establish that:

• \( \vec{MN} = \vec{PQ} \)
• \( \vec{MP} = \vec{NQ} \)

You conclude that MNPQ must indeed be a parallelogram.

This proof relies on recalling that parallelograms have opposite sides that are both parallel and equal. Consequently, using the properties of vector addition and geometry's structural elegance, you capture the essence of how these midpoint connections manifest as a parallelogram.