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Coordinate Geometry Formulas
Distance Formula Between Two Points
Explanation: Calculates the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\).
Formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Section Formula
Explanation: Determines the coordinates of a point dividing a line segment internally in the ratio \(m:n\).
Formula: \(\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)\)
Area of Triangle When Three Coordinates Given
Explanation: Calculates the area of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\).
Formula: \(\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|\)
Parallel Lines
Explanation: Lines with the same slope.
Formula: If \(m_1 = m_2\), the lines are parallel.
Perpendicular Lines
Explanation: Lines with slopes that are negative reciprocals.
Formula: If \(m_1 \cdot m_2 = -1\), the lines are perpendicular.
Slope of a Line
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Relation of Slope of Two Perpendicular and Parallel Lines
Explanation: Parallel lines have equal slopes; perpendicular lines have slopes that multiply to \(-1\).
Formula: Parallel: \(m_1 = m_2\), Perpendicular: \(m_1 \cdot m_2 = -1\)
Representation of Line
Explanation: General form of a line equation.
General equation: \(Ax + By + C = 0\)
Slope-intercept form: \(y=mx+c\)
Point slope form: \(y-y_0=m(x-x_0)\)
Two point form: \(y-y_1=\frac{(y_2-y_1)}{(x_2-x_1)}(x-x_1)\)
Slope intercept form: \(\frac{x}{a}+\frac{y}{b}=1\)
Equation of Two Perpendicular and Parallel Lines
Explanation: Equations for lines based on their slopes.
Formula: Parallel: \(y = mx + c\), Perpendicular: \(y = -\frac{1}{m}x + c\)
Equation of Perpendicular Line Passing Through a Point
Explanation: Finds the equation of a line perpendicular to a given line passing through a specific point \((x_1, y_1)\).
Formula: \(y - y_1 = -\frac{1}{m}(x - x_1)\)
Distance of a Line from a Given Point
Explanation: Calculates the perpendicular distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\).
Formula: \(d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\)
Angle Between Two Lines
If two lines have slopes m1 and m2, the angle theta between them is given by:
$$ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| $$
Sum Squared Error
Explanation: Measures the total deviation of predicted values from actual values.
Formula: \(SSE = \sum_{i=1}^n (y_i - \hat{y}_i)^2\)
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