Coordinate Geometry Apply

Answer 1

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates to represent and analyze geometric shapes and relationships on a Cartesian plane. Let's explore some key concepts and applications of coordinate geometry:

Basic Concepts

1. Coordinate System:

   • A Cartesian coordinate system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in the plane can be represented as an ordered pair \((x, y)\).

2. Distance Formula:

   • The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. Midpoint Formula:

   • The midpoint \(M\) of the segment connecting points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

4. Slope:

   • The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

5. Equation of a Line:

   • The equation of a line in slope-intercept form is: \[ y = mx + b \]
   • Here, \(m\) is the slope and \(b\) is the y-intercept.

Applications of Coordinate Geometry

1. Finding Equations of Lines:

   • Given two points, you can determine the slope and then use one point to find the line's equation.

2. Identifying Geometric Shapes:

   • You can use coordinates to define and identify geometric shapes like triangles, rectangles, and circles by analyzing their vertices and equations.

3. Intersection of Lines:

   • You can find the point of intersection between two lines by solving their equations simultaneously.

4. Area of Geometric Shapes:

   • You can calculate the area of triangles and other polygons using the coordinates of their vertices. For example, for a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\): \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

5. Transformation of Shapes:

   • You can use coordinate transformations to rotate, translate, or scale shapes.

Example Problems

1. Problem: Find the distance between the points (3, 4) and (7, 1).

   • Solution: \[ d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

2. Problem: Find the midpoint of the line segment connecting the points (-2, 3) and (4, -1).

   • Solution: \[ M = \left(\frac{-2 + 4}{2}, \frac{3 + (-1)}{2}\right) = \left(1, 1\right) \]

3. Problem: Determine the equation of the line that passes through the points (2, 3) and (4, 7).

   • Solution:
     • Calculate the slope: \[ m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \]
     • Using point-slope form \(y - y_1 = m(x - x_1)\) with point (2, 3): \[ y - 3 = 2(x - 2) \implies y - 3 = 2x - 4 \implies y = 2x - 1 \]

Conclusion

Coordinate geometry provides a powerful framework for solving problems involving geometric shapes and their relationships in a two-dimensional space. Applications range from simple distance calculations to complex transformations and equations of geometric figures. Understanding these concepts is essential for progress in geometry, algebra, and many practical fields such as physics, engineering, and computer graphics.