3-7 Coordinate Geometry - Grade 9 Math
Key Concepts
1. Coordinate Plane
The coordinate plane is a two-dimensional surface on which points are plotted using pairs of numbers called coordinates. It is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).
2. Distance Formula
The distance formula is used to find the distance between two points on the coordinate plane. For points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
3. Midpoint Formula
The midpoint formula is used to find the midpoint of a line segment on the coordinate plane. For endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((M)\) is given by \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
Detailed Explanation
Example 1: Using the Distance Formula
Points: \((3, 4)\) and \((7, 1)\)
Distance: \(d = \sqrt{(7 - 3)^2 + (1 - 4)^2}\)
\(d = \sqrt{4^2 + (-3)^2}\)
\(d = \sqrt{16 + 9}\)
\(d = \sqrt{25}\)
\(d = 5\)
Example 2: Using the Midpoint Formula
Endpoints: \((-2, 6)\) and \((4, -2)\)
Midpoint: \(M = \left(\frac{-2 + 4}{2}, \frac{6 + (-2)}{2}\right)\)
\(M = \left(\frac{2}{2}, \frac{4}{2}\right)\)
\(M = (1, 2)\)
Analogies for Clarity
Coordinate Plane as a Map
Think of the coordinate plane as a map where each point represents a location. The x-axis is like the longitude, and the y-axis is like the latitude. The distance formula helps you calculate the direct path between two locations, while the midpoint formula finds the halfway point between them.
Distance Formula as a Pathfinder
Imagine you are navigating through a city grid. The distance formula is like a pathfinder that calculates the shortest route between two points, considering all the streets and avenues.
Midpoint Formula as a Meeting Point
Consider the midpoint formula as finding the perfect meeting point between two friends. If each friend is at a different location on the map, the midpoint formula helps them find a spot that is equidistant from both their starting points.
Conclusion
Coordinate geometry is a powerful tool in mathematics, enabling us to locate points, measure distances, and find midpoints on a coordinate plane. By mastering the distance and midpoint formulas, you can solve a wide range of geometric problems and apply them to real-world scenarios.