Problem 76 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the line segments joining the midpoints of adjacent sides of any quadrilateral (four-sided polygon) form a parallelogram.

Step-by-Step Solution

Answer

The midpoints of a quadrilateral form a parallelogram by their consistent parallel and equal vectors.

Step 1: Identify Midpoints

Consider a quadrilateral with vertices labeled as \(A, B, C,\) and \(D\). Let's label the midpoints of these adjacent sides as \(M, N, O,\) and \(P\). Specifically, \(M\) is the midpoint of \(AB\), \(N\) is the midpoint of \(BC\), \(O\) is the midpoint of \(CD\), and \(P\) is the midpoint of \(DA\).

Step 2: Use Midpoint Formula

The coordinates of the midpoints can be calculated using the midpoint formula. For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint is given by \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Apply this formula to find the coordinates of \(M, N, O,\) and \(P\).

Step 3: Find Vectors Between Midpoints

Calculate the vectors for line segments \(MN, NO, OP,\) and \(PM\). These vectors should be based on the differences between the coordinate expressions of the midpoints. For example, the vector \(\overrightarrow{MN}\) would be \((N_x - M_x, N_y - M_y)\).

Step 4: Verify Parallel and Equal Vectors

Show that opposite vectors \(\overrightarrow{MN}\) and \(\overrightarrow{OP}\) are equal, and \(\overrightarrow{NO}\) and \(\overrightarrow{PM}\) are equal. This means that both pairs must have the same magnitude and direction to confirm they are parallel, confirming the shape as a parallelogram.

Key Concepts

Midpoint Formula

The midpoint formula is a crucial tool in geometry. It allows us to find the center point between two given points on a Cartesian plane. It uses the average of the x-coordinates and the y-coordinates of the endpoints to determine this midpoint. For points

• the formula is written as: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

Applying this formula is straightforward. Just take each pair of coordinates, add them, and split it by two. Repeat the process for both x and y coordinates. The result gives you the coordinates of the midpoint, useful to form line segments in shapes like quadrilaterals.

Vectors

Vectors are used to represent direction and magnitude. In geometry, they help us describe line segments and their properties. A vector from point A to point B is usually denoted as \(\overrightarrow{AB}\). It consists of two parts:

• Magnitude: The length of the vector
• Direction: The direction from A to B

To find the vector between two points, subtract the coordinates of the starting point from the ending point's coordinates. For example, if A is

• the vector from A to B is:\[(B_x - A_x, B_y - A_y)\]

Vectors are helpful in proving geometrical assertions, like determining if two sides of a shape are parallel, a key aspect when showing a shape is a parallelogram.

Quadrilateral

A quadrilateral is a polygon with four sides and four vertices. It can take various forms, such as squares, rectangles, and, more generally, any four-sided shape. Each corner point is known as a vertex. Quadrilaterals are categorized based on the relationships between sides and angles, such as:

• Parallelograms where opposite sides are parallel
• Trapezoids with one pair of parallel sides

Identifying the properties such as midpoint coordinates and vectors in a quadrilateral is essential for solving problems like ensuring midpoints form another geometric shape.

Parallelogram

A parallelogram is a special type of quadrilateral. The defining feature is that its opposite sides are both parallel and equal in length. This means:

• Opposite sides have the same length
• Each pair of opposite angles are equal

When you draw lines connecting midpoints of adjacent sides in any quadrilateral, a parallelogram forms, showing the elegant symmetry these shapes can offer. Verifying parallelism involves using vectors to prove their equality in magnitude and direction, ensuring the segments connecting the midpoints truly form a parallelogram.