91Ó°ÊÓ

Coordinate geometry is a branch of mathematics that uses a coordinate system to study the position of points, lines, and various geometric shapes on a plane. Here, each point in the plane is described using ordered pairs, often denoted as \((x, y)\). In the context of our problem, the plane is the usual two-dimensional Cartesian plane. Each point's position is determined by its coordinates relative to two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
Understanding coordinate geometry is essential for calculating distances, angles, and other properties of geometric figures on the plane. Here are some basics to remember:

• The x-coordinate of a point tells you how far along the horizontal axis the point is located.
• The y-coordinate indicates how far along the vertical axis the point is.
• The origin, denoted as \((0, 0)\), is where the axes intersect.

In problems like calculating the distance between two points, coordinate geometry helps us systematically figure out how far apart these points are positioned on the plane.

To find the distance between two points in a plane, we use the distance formula. This formula is derived from the basic principles of coordinate geometry and helps to calculate the straight-line distance, or Euclidean distance, between two points. The distance formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where:

• \( (x_1, y_1) \) are the coordinates of the first point.
• \( (x_2, y_2) \) are the coordinates of the second point.

Let's apply the distance formula to our specific example, points \( T(6,4) \) and \( U(2,2) \):

• Calculate the differences: \( x_2 - x_1 = 2 - 6 = -4 \) and \( y_2 - y_1 = 2 - 4 = -2 \).
• Square these differences: \((-4)^2 = 16\) and \((-2)^2 = 4\).
• Sum up the squares: \(16 + 4 = 20\).
• Finally, take the square root: \(\sqrt{20} \approx 4.4721\), and rounding this gives approximately \(4.5\).

By following these steps, you can find the distance between any two points on a plane.

The distance formula is closely related to the Pythagorean theorem, a fundamental principle in geometry. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Using triangle terminology:
\[ c^2 = a^2 + b^2 \]
Where:

• \(c\) is the hypotenuse.
• \(a\) and \(b\) are the triangle's other two sides.

When applied to finding the distance between points \((x_1, y_1)\) and \((x_2, y_2)\), think of the distance between the points as the hypotenuse of a right triangle. The differences \(x_2 - x_1\) and \(y_2 - y_1\) represent the lengths of the triangle's other two sides.

• Square each difference, which represents the squares of \(a\) and \(b\).
• Add both squares to find \(c^2\), the square of the hypotenuse.
• The square root of \(c^2\) is \(c\), our desired distance.

Thus, the Pythagorean theorem not only helps in geometry but also anchors the distance formula, giving it a clear mathematical foundation.