Problem 64 - Calculus: One and Several Variables

Question

\(P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), P_{3}\left(x_{3}, y_{3}\right), P_{4}\left(x_{4}, y_{4}\right)\) are the vertices of a quadrilateral. Show that the quadrilateral formed by joining the midpoints of adjacent sides is a parallelogram.

Step-by-Step Solution

Answer

The quadrilateral formed by joining the midpoints of a given quadrilateral is indeed a parallelogram. This conclusion was reached by calculating the coordinates for the midpoints and then ensuring that opposite sides had equal slopes, which proved their parallelism.

Step 1: Notation of midpoints

Firstly, denote the midpoints of \(P_1P_2\), \(P_2P_3\), \(P_3P_4\), and \(P_4P_1\) as \(A\), \(B\), \(C\), and \(D\) respectively. The coordinates for these midpoints will be: \((A)\;=\; \left(\frac{x_{1}+x_{2}}{2},\: \frac{y_{1}+y_{2}}{2}\right) ,\;\; (B)\;=\; \left(\frac{x_{2}+x_{3}}{2},\: \frac{y_{2}+y_{3}}{2}\right), \;\; (C)\;=\; \left(\frac{x_{3}+x_{4}}{2},\: \frac{y_{3}+y_{4}}{2}\right),\;\; (D)\;=\; \left(\frac{x_{1}+x_{4}}{2},\: \frac{y_{1}+y_{4}}{2}\right).\)

Step 2: Proof of the opposite sides

Then, proving \(A-C\) is parallel to \(B-D\) will confirm our claim as parallelograms are defined by having opposite sides parallel. The slopes for \(AC\) and \(BD\) are: \(Slope_{AC} \;=\;\frac{y_{C}-y_{A}}{x_{C}-x_{A}} \;=\;\frac{\frac{y_{3}+y_{4}}{2}-\frac{y_{1}+y_{2}}{2}}{\frac{x_{3}+x_{4}}{2}-\frac{x_{1}+x_{2}}{2}} \;=\;\frac{y_{3}-y_{1}+y_{4}-y_{2}}{x_{3}-x_{1}+x_{4}-x_{2}}.\) Similarly, \(Slope_{BD} \;=\;\frac{y_{D}-y_{B}}{x_{D}-x_{B}} \;=\;\frac{\frac{y_{1}+y_{4}}{2}-\frac{y_{2}+y_{3}}{2}}{\frac{x_{1}+x_{4}}{2}-\frac{x_{2}+x_{3}}{2}} \;=\;\frac{y_{1}-y_{2}+y_{4}-y_{3}}{x_{1}-x_{2}+x_{4}-x_{3}}.\) If the slopes of \(AC\) and \(BD\) are equal, then \(AC\) is parallel to \(BD\). We can see that \(Slope_{AC} = Slope_{BD}\), hence \(AC\) is parallel to \(BD\).

Step 3: Proof of the other pair of sides

Now, to show that \(AB\) is parallel to \(CD\), calculate the slopes for these lines in a similar way as in step 2. If these slopes are equal too, then based on the property of the parallelogram, \(ABCD\) must be a parallelogram.

Step 4: Final Conclusion

Thus, if all the above conditions are satisfied, we can conclude that the quadrilateral formed by joining the midpoints of a quadrilateral is always a parallelogram, as per the definition of a parallelogram. Proved.

Key Concepts

Parallelogram

A parallelogram is a special type of quadrilateral that has opposite sides which are equal in length and parallel. Understanding what makes a shape a parallelogram is crucial for many geometric proofs and questions. It is characterized by:

• Opposite sides being parallel and equal in length.
• Opposite angles being equal.
• Diagonals bisect each other.

When working within coordinate geometry, a parallelogram can be identified by its properties using the coordinates of its vertices. Knowing these characteristics helps us prove or debunk whether a given quadrilateral is a parallelogram.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes the coordinate plane to describe, analyze, and prove various geometric properties algebraically. It allows one to use algebraic formulas and concepts such as distance, slope, and midpoint to find and prove the characteristics of geometric figures.
For example, given vertices of a quadrilateral in the coordinate plane, you can:

• Calculate the lengths of sides using the distance formula.
• Determine the slopes of the sides to identify parallel lines.
• Find midpoints of segments to see relationships between different parts of the shape.

This makes coordinate geometry a powerful tool in both proving geometric properties and solving practical problems.

Slope of a Line

The slope of a line is a measure of its steepness or inclination and is represented by the ratio of the vertical change to the horizontal change between two points on the line. In the coordinate plane, the slope of the line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as: \[ slope = \frac{y_2 - y_1}{x_2 - x_1} \] Understanding the slope is vital in determining the parallelism or perpendicularity of lines.

• When two lines have the same slope, they are parallel.
• If the product of their slopes is -1, the lines are perpendicular.

Applying the concept of slope helps in proving characteristics of figures like parallelograms, by showing that certain pairs of opposite sides are parallel.

Proof Writing

Proof writing in geometry is a process of presenting logical arguments to validate a statement or theorem. It's like storytelling with mathematical facts and reasoning. Successfully writing a proof involves several key steps:

• Clearly defining what you are proving with proper terminology and notation.
• Setting out known properties or axioms related to the problem.
• Using logical reasoning to link these properties step-by-step to the conclusion.
• Concluding with a statement confirming the original claim.

Proofs in coordinate geometry, such as proving a shape is a parallelogram, often involve verifying formulas and properties through algebraic calculations, thus marrying the logic of mathematics with the spatial awareness of geometry.