Coordinate Geometry
In this book, you will learn the following: [list] [*]find the coordinates of the midpoint of a line segment; [*]compute the gradient of a line that is parallel or perpendicular to a given line; [*]calculate the area of a rectilinear figure given the coordinates of its vertices using the shoelace method; [*]obtain the equation of a circle given its centre and radius, and vice versa. [/list]
Table of Contents
- Midpoints
- Formula for mid-value
- Midpoint of a Line Segment (or Two Coordinates)
- Practice: Midpoint of Line Segment
- Practice 2: Midpoint of Two Coordinates
- Midpoint Quiz
- Parallelogram Coordinate Geometry
- Parallelogram Midpoint Practice
- Parallel and Perpendicular Lines
- Gradient Quiz
- Angle of Inclination
- parallel lines, gradient, collinearity
- perpendicular lines
- Practice: Parallel & Perp lines
- Gradients of Parallel and Perpendicular Lines Quiz
- Equation of Straight Line
- Equation of Straight Line Quiz
- Equation of Straight Line
- Equation of Perpendicular Bisector
- Area of a Polygon
- Area of a Polygon
- Equation of Circles
- Circle Equation
- Practice: drawing a circle to match equation
- Practice: Equation of Circles
- Circle Equation: General Form
- Practice: Circle Eqn General Form, Radius and Center
- Eqn of Circles Quiz
- Circle: Radius & Tangent
- Practice: Eqn tangent to Circle
- Perpendicular Bisector of Chord
- Finding Equation of Circle
Formula for mid-value
In this segment, we will learn how to find the midpoint of a line segment in the cartesian coordinate plane. Or, in simpler terms, given two coordinates, how we will find the coordinates of the midpoint.[br][br]We will first start off with something that makes better sense for us - finding the middle value of two values.
Mid-value of Two Numbers
Gradient Quiz
Equation of Straight Line Quiz
Area of a Polygon
Using the Shoelace Formula to calculate area of a polygon
As a summary, given the vertices of the polygon [math]P_1(x_1,y_1)[/math], [math]P_2(x_2,y_2)[/math] ,[math]\cdots[/math], [math]P_n(x_n,y_n)[/math], the area of a polygon is given by this formula:[br][center][math]\text{Area of a Polygon}=\frac{1}{2}\begin{vmatrix}x_1 & x_2 & x_3 & \cdots & x_n & x_1 \\y_1 & y_2 & y_3 & \cdots & y_n & y_1 \end{vmatrix}[/math][/center]where the products taken in the direction[math]\searrow[/math]are given [color=#38761d][b]positive [/b][/color]signs, [br]and products taken in the direction[math]\nearrow[/math]are given [b][color=#cc0000]negative [/color][/b]signs.[br][br]E.g. for a polygon with three vertices [math]P_1(x_1,y_1),P_2(x_2,y_2),P_3(x_3,y_3)[/math], the area of the polygon is [center][math]\begin{align}\text{Area of a Polygon}&=\frac{1}{2}\begin{vmatrix}x_1 & x_2 & x_3 & x_1 \\y_1 & y_2 & y_3 & y_1 \end{vmatrix}\\ &= \frac{1}{2}x_1 y_2+x_2 y_3 + x_3 y_1 - \left(x_2 y_1 + x_3 y_2 + x_1 y_3\right)\end{align}[/math][/center][b]NOTE[/b]: this formula only works fine when the vertices are considered in a [b][u]counterclockwise direction[/u][/b].
Quiz Yourself!
Circle Equation
Equation of Circle in Standard Form
In this section, we will learn about the equation of circles in standard form.[br][br][center][math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math][/center]In the applet below, play around with the sliders h, k, and r. [br]Observe how each of these variables control different aspects of the circle.
What does the variable h in [math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math] control?
What does the variable k in [math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math] control?
What does the variable r in [math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math] control?