Problem 63 - Calculus: One and Several Variables

Question

Prove that cach diagonal of a parallelogram bisccts the other. HINT: Introduce a coordinate systcm with one vertex al the origin and one side on the positive \(x\) -axis.

Step-by-Step Solution

Answer

By setting the vertices of the parallelogram in a coordinate system, it's shown that when the x-coordinate of \(B\) is half the x-coordinate of \(C\), the midpoints of both diagonals coincide. Hence, each diagonal of a parallelogram bisects the other.

Step 1: Set up the coordinate system

Let's identify the four vertices of the parallelogram as \(A\), \(B\), \(C\), and \(D\). We can set \(A\) at the origin \((0,0)\), \(B\) on the positive x-axis at point \((a,0)\), \(C\) at point \((b,c)\) and \(D\) at point \((a,c)\).

Step 2: Find the midpoint of the diagonals

The midpoint of a line segment joining points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)\). Using this equation, we can find the midpoint of both the diagonals \(\overline{AC}\) and \(\overline{BD}\). The midpoint of \(\overline{AC}\) is \(\left( \frac{0+b}{2}, \frac{0+c}{2} \right) = \left(\frac{b}{2}, \frac{c}{2} \right) \) and the midpoint of \(\overline{BD}\) is \( \left( \frac{a+a}{2}, \frac{0+c}{2} \right) = \left(\frac{2a}{2}, \frac{c}{2} \right) = \left(a, \frac{c}{2} \right) \)

Step 3: Compare the midpoints

Clearly, when \(a = \frac{b}{2}\), the midpoints of \(\overline{AC}\) and \(\overline{BD}\) are the same, which means these diagonals bisect each other.

Key Concepts

Coordinate Geometry and its Application to Parallelograms

Coordinate geometry, also known as analytical geometry, is the study of geometry using a coordinate system. This innovative approach combines algebra and geometry to solve problems involving shapes, sizes, and positions of figures on a plane. One classic problem solved using coordinate geometry is proving that the diagonals of a parallelogram bisect each other.

In the context of our exercise, a parallelogram is placed on a coordinate system such that one vertex is at the origin, (0,0), and one side lies along the X-axis. By positioning a geometric figure within a coordinate system, we can easily use algebra to analyze its properties. This facilitation is particularly useful because it allows for the application of formulas and theorems to find distances, midpoints, and other geometric features. Furthermore, when the vertices of the parallelogram have known coordinates, the principles of coordinate geometry enable us to apply concrete numerical methods to abstract geometric problems, creating pathways to unequivocal proofs.

The Power of Geometric Proof

A geometric proof is a series of logical deductions used to confirm a particular geometric statement. These proofs employ definitions, theorems, and axioms in order to reach a conclusion, or 'prove' a given hypothesis. In proving that a parallelogram's diagonals bisect each other, the geometric proof serves as a precise and rigorous method to show that upon drawing, the diagonals indeed intersect each other exactly in the middle.

The beauty of a geometric proof lies in its requirement for a clear logical sequence. Each step is both necessary and sufficient, leading one to the final truth of the matter. Our problem at hand is resolved through the application of a geometric proof which utilizes coordinate geometry and properties known about parallelograms. This combination of specific examples and broad principles makes geometric proof not only elegant but also universally applicable across diverse problems.

Midpoint Formula: Bridging the Diagonals of Parallelograms

The midpoint formula is a critical tool in coordinate geometry and plays a central role in our parallelogram proof. It states that the midpoint, 'M', of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). This formula is the arithmetic mean of the endpoints' coordinates and gives us the exact center point of a line segment.

By applying the midpoint formula to the diagonals of the parallelogram, we can substantiate the theorem that the diagonals bisect each other. For our specific parallelogram, with the diagonals \(\overline{AC}\) and \(\overline{BD}\), the midpoints are calculated using the coordinates of their endpoints. When the calculated midpoints are identical, it proves that each diagonal cuts the other in half. The midpoint formula is a simple yet profound expression that transforms this geometric assertion into an algebraic check, providing a concrete foothold for verification.