Problem 26 - The Calculus with Analytic Geometry

Question

Prove analytically that the sum of the squares of the distances of any point from two opposite vertices of any rectangle is equal to the sum of the squares of its distances from the other two vertices.

Step-by-Step Solution

Answer

The sum of the squares of the distances from any point to two opposite vertices is equal to the sum of the squares of the distances to the other two vertices.

Step 1: Define Coordinates

Let the rectangle be defined by the vertices at coordinates \(A(0, 0)\), \(B(a, 0)\), \(C(a, b)\), and \(D(0, b)\). Let \(P(x, y)\) be any point in the plane.

Step 2: Calculate Distance to Opposite Vertices

Calculate the square of the distances from point P to the vertices \(A(0,0)\) and \(C(a,b)\). These distances are given by: \(PA^2 = x^2 + y^2\) and \(PC^2 = (x-a)^2 + (y-b)^2\).

Step 3: Calculate Distance to the Other Pair of Vertices

Next, calculate the square of the distances from point P to the vertices \(B(a,0)\) and \(D(0,b)\). These distances are given by: \(PB^2 = (x-a)^2 + y^2\) and \(PD^2 = x^2 + (y-b)^2\).

Step 4: Express Sum of Squares for Both Sets

Sum the squares of distances to the opposite vertices: \(PA^2 + PC^2 = x^2 + y^2 + (x-a)^2 + (y-b)^2\). Sum the squares of distances to the other two vertices: \(PB^2 + PD^2 = (x-a)^2 + y^2 + x^2 + (y-b)^2\).

Step 5: Simplify and Compare Sums

Expand the expressions and combine like terms for both sums: \(PA^2 + PC^2 = x^2 + y^2 + x^2 - 2ax + a^2 + y^2 - 2by + b^2\) and \(PB^2 + PD^2 = x^2 - 2ax + a^2 + y^2 + x^2 + y^2 - 2by + b^2\). Both expressions simplify to: \[PA^2 + PC^2 = PB^2 + PD^2 = 2x^2 + 2y^2 + a^2 + b^2 - 2ax - 2by\].

Step 6: Conclude the Proof

Since the simplified sums are equal, it is shown that the sum of the squares of the distances from any point to two opposite vertices is equal to the sum of the squares of the distances to the other two vertices.

Key Concepts

Analytical Geometry

Analytical geometry, also known as coordinate geometry, integrates algebra and geometry by using a coordinate system. It allows us to represent geometric shapes and figures with algebraic equations. In this problem, we identify the coordinates of the vertices of the rectangle and an arbitrary point P in the plane.

Using coordinates, we convert geometric situations into algebraic equations. For example, we defined the rectangle's vertices as A(0,0), B(a,0), C(a,b), and D(0,b), making it easier to calculate distances and sum the squares of those distances effectively.

Distance Formula

The distance formula is crucial in analytical geometry. It's a way to determine the distance between two points in a plane using their coordinates. The formula stems from the Pythagorean theorem and is written as: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

To find the square of the distance between points P(x,y) and a vertex like A(0,0), we simplify the formula by omitting the square root, leading to \(PA^2 = x^2 + y^2\). Similarly, for point C(a,b), the squared distance is \(PC^2 = (x-a)^2 + (y-b)^2\). This method ensures efficient calculations for summing distances later in our proof.

Coordinate Geometry

Coordinate geometry is powerful for solving problems with geometric shapes, such as this rectangle. It involves plotting geometric elements on a coordinate plane, using algebra to explore their relationships.

By plotting the vertices and using coordinates, we simplify and visualize geometric problems. When we consider distances from an arbitrary point P to the vertices of the rectangle, the use of exact coordinates streamlines our calculations and proofs.

Proof Techniques

Proof techniques in coordinate geometry often involve algebraic manipulations and logical steps. Here, we've tackled the problem by following a structured approach:

• We defined coordinates for the vertices and the arbitrary point.
• We calculated squared distances from the point to each vertex.
• We summed and simplified these squared distances.
• Finally, we compared these simplified expressions to conclude the proof.

Following a step-by-step method ensures clarity and correctness. It also helps in identifying and addressing any potential errors in the proof.