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Coordinate geometry, also known as analytic geometry, allows us to represent geometric shapes in a numerical way by using the Cartesian coordinate system. Each point on the plane is represented by a pair of numbers corresponding to its horizontal and vertical positions, known as coordinates. The beauty of coordinate geometry lies in its ability to convert geometric problems into algebraic ones, making them easier to solve.

For example, in our rectangle problem, by ascribing coordinates to the rectangle’s vertices, we can utilize algebraic methods to prove properties that would be difficult to demonstrate visually or through traditional geometric proofs. This is particularly useful for proving congruency, parallel lines, and other geometric properties without the need for constructing angles or measuring lengths directly on a shape.

The distance formula is a tool derived from the Pythagorean theorem used in coordinate geometry to calculate the distance between any two points in the Cartesian plane. The formula is expressed as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This allows us to determine lengths in a geometric figure by simply knowing the coordinates. In the context of our rectangle problem, the distance formula is utilized to calculate the lengths of diagonals AC and BD, allowing us to compare them algebraically. As we see in our step-by-step solution, once we have the coordinates of the ends of the diagonals, we can easily find their lengths. The results demonstrate that the diagonals are congruent since they share the same length, a vital part of the proof we're seeking.

Congruent segments are simply line segments that are equal in length. In geometry, congruence is fundamental, as it indicates that segments, angles, or shapes are identical in size and shape. When working with coordinate geometry, congruence can be proven by showing that the distance between the endpoints of two segments is the same.

In our exercise, proving that the diagonals of a rectangle are congruent involves showing that the lengths of the diagonals are identical. With the help of the distance formula, we determined that segments AC and BD have equal lengths. Therefore, they are congruent segments. This concept is vital; without proving the congruence of diagonals, we cannot fully understand the rectangle's properties.

Rectangles are four-sided polygons with some very specific and useful properties: all angles are right angles, opposite sides are parallel and equal in length, and the diagonals bisect each other and are congruent.

When it comes to proofs using coordinate geometry, the rectangle's properties can be systematically demonstrated using algebra. This is performed by calculating distances for sides and diagonals and showing that the right angle exists by confirming the slope of perpendicular lines. Our exercise capitalizes on these properties to prove that the diagonals of a rectangle are congruent—a key characteristic defining the rectangle. Understanding these properties helps in solving not only this problem but also in recognizing patterns and solving more complex geometrical problems involving rectangles.