91Ó°ÊÓ

In the realm of mathematics, coordinate geometry is a powerful tool that merges algebra and geometry, enabling us to precisely locate points, lines, and shapes in a plane using numerical coordinates. The two points in our exercise, (9.5,-2.6) and (-3.9,8.2), lie on a two-dimensional plane commonly referred to as the Cartesian plane. This plane is divided into four quadrants by a horizontal axis (the x-axis) and a vertical axis (the y-axis).

Each point has a pair of coordinates: the first number represents the position along the x-axis, and the second number represents the position along the y-axis. By plotting these points on the Cartesian plane, we can visualize their location and the line segment that connects them; this segment represents the 'distance' we are trying to calculate.

Understanding how to find the distance between two points is fundamental in coordinate geometry. It's like having a mathematical ruler that measures the straight-line distance from one point to another across the Cartesian plane. The formula in our solution, \(D = \sqrt{(B_x-A_x)^2 + (B_y-A_y)^2}\), is derived directly from the Pythagorean theorem, which we'll discuss further in the next section.

The distance formula calculates the length of the hypotenuse of a right-angled triangle, the sides of which are parallel to the axes and represent the horizontal and vertical distances between the points. Our exercise determines this distance by substituting the specific coordinates of points A (9.5, -2.6) and B (-3.9, 8.2) into the formula, thus making what might seem complex at first glance, a straightforward arithmetic problem.

The Pythagorean theorem, a cornerstone of geometry, tells us that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is beautifully simple and universally written as \(a^2 + b^2 = c^2\).

In the context of our exercise, if we imagine or draw a right-angled triangle with points A and B as two corners and the right angle located at the point where lines drawn from A and B parallel to the axes intersect, the distance we want to find is actually the hypotenuse ‘c’. Here, ‘a’ and ‘b’ are the changes in the x and y coordinates respectively, and by plugging these into the Pythagorean theorem, we can solve for ‘c’, which is the distance formula we've used.