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Coordinate geometry is a branch of mathematics that uses a coordinate system to represent and reason about geometric figures. When you have a set of points on a plane, you can represent them using coordinates. A coordinate system involves two number lines that are perpendicular, called the x-axis and y-axis, intersecting at a point called the origin.
The points in the system are denoted by ordered pairs \(x, y\), where:

• x: represents the horizontal position of the point, called the x-coordinate.
• y: represents the vertical position of the point, called the y-coordinate.

In the exercise, the points A and B have coordinates (\(-12, -3\)) and (\(-5, -7\)) respectively. These coordinates tell us how far each point is from the origin and from each other. Understanding the placement of these points is essential in calculating distances and understanding their relation to one another on the plane. Coordinate geometry provides valuable tools to easily calculate distances between points, identifies geometric shapes, and solves many real-world problems involving spatial relationships.

The distance formula is a useful application of coordinate geometry used to calculate the distance between two points on a Cartesian plane. The formula is derived from the Pythagorean theorem, which relates the lengths of the sides of a right triangle to the length of the hypotenuse.
The formula is given by \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Simply put:

• d: represents the distance between the two points.
• (x_1, y_1): are the coordinates of the first point.
• (x_2, y_2): are the coordinates of the second point.

By substituting the given coordinates into the formula, we compute how far Point A \( (-12, -3) \) is from Point B \( (-5, -7) \). The differences \((-5 + 12)\) and \((-7 + 3)\) are obtained to ultimately solve for the distance. This step is crucial for spatial questions, aiding in the visual and quantitative analysis of geographical data. Whether you are just calculating simple point-to-point distances on a map or analyzing complex spatial datasets, the distance calculation is an indispensable skill.

Squaring numbers is a fundamental mathematical operation that involves multiplying a number by itself. This concept is important in distance calculation where differences in coordinates are squared.
Squaring has several key properties:

• Squaring a positive number or a negative number gives a positive result because multiplying two numbers of the same sign results in a positive number.
• Squaring 0 results in 0, as 0 multiplied by any number is 0.

In our exercise, after calculating the differences \(7\) and \(-4\), these values are squared to obtain \(7^2 = 49\) and \((-4)^2 = 16\). Adding these squares gives the intermediary step \(65\), before taking the square root to find the final distance. Remember that squaring numbers simplifies expressions and is a common step in various geometric and algebraic computations. This preparation helps reinforce both understanding and confidence in using mathematical formulas correctly.