AOS1 Topic 1: Coordinate Geometry (Part 2)
Coordinate geometry (or analytic geometry) is defined as the study of geometry using the coordinate points. Using coordinate geometry, it is possible to find the distance between two points, dividing lines in m:n ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc.
Coordinate Geometry Terms
Cartesian Coordinate System:
Consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where the axes intersect is called the origin, denoted as \((0, 0)\). Any point in the plane can be described by an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance from the origin.
Distance between two points
The distance between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Midpoint of a line segment
The midpoint of the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is the point with coordinates:
\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Gradient of a straight line
The gradient \(m\) of a straight line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Equation of a straight line
Gradient-intercept form: A straight line with gradient \(m\) and y-axis intercept \(c\) has the equation:
\[ y = mx + c \]
Point-gradient form: The equation of a straight line passing through a given point \( (x_1, y_1) \) and having gradient \(m\) is:
\[ y - y_1 = m(x - x_1) \]
Two-point form: The equation of a straight line passing through two given points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ y - y_1 = m(x - x_1) \]
where \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Intercept form: The straight line passing through the two points \( (a, 0) \) and \( (0, b) \) has the equation:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
Tangent of the angle of slope
For a straight line with gradient \(m\), the angle of slope is found using:
\[ m = \tan \theta \]
where \( \theta \) is the angle that the line makes with the positive direction of the x-axis.
Perpendicular straight lines
If two straight lines are perpendicular to each other, the product of their gradients is \(-1\), i.e., \( m_1 m_2 = -1 \) (unless one line is vertical and the other horizontal).
Coordinate geometry is a powerful tool for solving geometric problems and proving geometric theorems using algebraic methods. It bridges the gap between algebra and geometry, providing a deeper understanding of both subjects.
What is a Co-ordinate and a Co-ordinate Plane?
You must be familiar with plotting graphs on a plane, from the tables of numbers for both linear and non-linear equations. The number line which is also known as a Cartesian plane is divided into four quadrants by two axes perpendicular to each other, labelled as the x-axis (horizontal line) and the y-axis (vertical line).
The four quadrants along with their respective values are represented in the graph below:
Quadrant 1: (+x, +y)
Quadrant 2: (-x, +y)
Quadrant 3: (-x, -y)
Quadrant 4: (+x, -y)
The point at which the axes intersect is known as the origin. The location of any point on a plane is expressed by a pair of values (x, y) and these pairs are known as the coordinates.
Coordinate Geometry Theorems
Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points
Consider the same points A and B, which have coordinates \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Let M(x,y) be the midpoint of lying on the line connecting these two points A and B. The coordinates of point M is given as:
\[ M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Angle Formula: To Find The Angle Between Two Lines
Consider two lines A and B, having their slopes \(m_1\) and \(m_2\), respectively. Let “θ” be the angle between these two lines, then the angle between them can be represented as:
\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
Special Cases:
Case 1: When the two lines are parallel to each other,
\(m_1 = m_2 = m\)
Substituting the value in the equation above,
\[ \tan(\theta) = \left| \frac{m - m}{1 + m \cdot m} \right| = 0 \]
Case 2: When the two lines are perpendicular to each other,
\(m_1 \cdot m_2 = -1\)
Substituting the value in the original equation,
\[ \tan(\theta) = \left| \frac{-1 - m_1 m_2}{1 + m_1 m_2} \right| = \text{undefined} \]
\( \Rightarrow \theta = 90^\circ \)
Section Formula: To Find a Point Which Divides a Line into m:n Ratio
Consider a line A and B having coordinates \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Let P be a point that divides the line in the ratio m:n, then the coordinates of the point P is given as:
When the ratio m:n is internal:
\[ P \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
When the ratio m:n is external:
\[ P \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right) \]
Students can follow the link provided to learn more about the section formula along its proof and solved examples.
Area of a Triangle in Cartesian Plane
The area of a triangle in coordinate geometry whose vertices are \((x_1, y_1)\), \((x_2, y_2)\) and \((x_3, y_3)\) is:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
If the area of a triangle whose vertices are \((x_1, y_1)\), \((x_2, y_2)\) and \((x_3, y_3)\) is zero, then the three points are collinear.