91Ó°ÊÓ

Understanding the coordinates of points is essential for solving various geometrical problems. The Cartesian coordinate system represents a point's position with a pair of numerical values, which are systematically laid out in a two-dimensional space. In our given exercise, the two points are represented as \( (9.5, -2.6), (-3.9, 8.2) \) where each pair consists of an x-coordinate (horizontal positioning) and a y-coordinate (vertical positioning).

The order of these coordinates is crucial, with the first number denoting the x-coordinate and the second denoting the y-coordinate. It tells us exactly where a point sits on the plane. By using a grid, we imagine the first point \( (9.5, -2.6) \) to be 9.5 units to the right of the origin along the x-axis and 2.6 units down along the y-axis. Conversely, the second point \( (-3.9, 8.2) \) is 3.9 units to the left and 8.2 units up from the origin.

The square root calculation is a mathematical operation that determines the quantity which, when multiplied by itself, will give the original number. For instance, the square root of 9 is 3 because 3 times 3 is 9. The symbol for square root is \( \sqrt{} \).

In the context of the Distance Formula, once we square the differences of the coordinates and add them together, we're left with the value under the square root that gives us the distance between the points. In our exercise, this calculation is \( \sqrt{296.2} \). Finding the square root of a non-square number without a calculator can be done using estimation or series expansion methods; however, precise calculations are often done with a calculator. This result will provide us with the linear distance between the two points in a two-dimensional space.

The Distance Formula is a derivation from the Pythagorean Theorem applied in a coordinate system to calculate the straight-line distance between two points. It states that the distance \( d \) between two points with the coordinates \( (x1, y1) \) and \( (x2, y2) \) is expressed as: \[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

In our exercise, the formula is applied as follows: we subtract the x-coordinate of the first point from the x-coordinate of the second point and square the result. We repeat this process for the y-coordinates. Then, add these squared differences to get the square of the distance.
To put the application in practice: \[ d = \sqrt{{(-3.9 - 9.5)}^2 + {(8.2 - (-2.6))}^2} \] After simplifying, the intermediate calculations yield \( \sqrt{179.56 + 116.64} \) which then simplifies further to \( \sqrt{296.2} \), thus determining the length of the line segment that connects the two points directly.