91Ó°ÊÓ

Coordinate geometry, also known as analytic geometry, is the study of geometric properties and spatial relations using algebraic equations. It involves the use of coordinate points to represent geometric figures. For instance, in our exercise, we analyzed the points \( (c, c-d) \) and \( (d, c+d) \), which are expressed in coordinate form, to determine their distance apart. The importance of coordinate geometry lies in its ability to describe positions and relationships precisely in a plane. It uses a system of coordinates where each point is defined by a pair (or triplet in 3D) of numbers.

The distance formula is a fundamental concept in coordinate geometry. It is used to calculate the distance between two points in a plane, given their coordinates. The distance formula is derived from the Pythagorean theorem and is given by: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This equation essentially finds the hypotenuse of a right triangle formed by the vertical and horizontal distances between two points. In our exercise, we substituted the specific coordinates of points \( (c, c-d) \) and \( (d, c+d) \) into the formula, leading us through several simplification steps to find the final distance, showing how the algebraic manipulation of coordinate differences gives the desired measure of separation.

Algebraic manipulation is the process of using algebraic techniques to simplify, solve, or rewrite expressions and equations. It's a critical part of solving geometry problems, like finding the distance between points, where coordinates involve variables. In the exercise, starting from the distance formula, we substituted and manipulated the coordinates: \[ \text{Distance} = \sqrt{(d - c)^2 + ((c+d) - (c-d))^2} \] Then simplified step-by-step: \[ \text{Distance} = \sqrt{(d - c)^2 + (2d)^2} \] \[ \text{Distance} = \sqrt{(d-c)^2 + 4d^2} \] Next, we expanded and combined like terms: \[ \text{Distance} = \sqrt{5d^2 - 2cd + c^2} \] This step-wise simplification shows how algebraic techniques are used to reach a final, simplified form.