Distance between two points :
Distance between two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is
Mid-point of line segment :
The mid-point M, of the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is
Section Formula (Internal Division) :
Let A(x_{1}, y_{1}) and B(x_{2}, y_{2}) be two distinct points such that point p(x, y) divides AB internally in the ratio m : n.
Then the coordinates of P are given by
Section Formula (Internal Division) :
Let A(x_{1}, y_{1}) and B(x_{2}, y_{2}) be two distinct points such that point p(x, y) divides AB externally in the ratio m : n.
Then the coordinates of P are given by
Centroid of a Triangle :
The coordinates of the centroid (G) of a triangle with vertices A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are given by
Area of a Triangle :
Area of a Triangle :
Let A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) be the vertices of a triangle.
Then the area of ΔABC is the absolute value of the expression :
The vertices A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) of ΔABC are said to be “taken in order” if A, B, C are taken in counter-clock wise direction. If we do this, then area of DABC will never be negative.
Collinearity of three points :
Let A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) be the three distinct points. If these three points are collinear (lie on the same straight line), then ΔABC = 0.
That is,
Area of a Quadrilateral :
Let A(x_{1}, y_{1}), B(x_{2}, y_{2}), C(x_{3}, y_{3}) and D(x_{3}, y_{3}) be the vertices of a quadrilateral.
Then the area of quadrilateral ABCD is
Slope of a straight line :
If θ is the angle of inclination of a non-vertical straight line, then tanθ is called the slope or gradient of the line and is denoted by m.
Therefore the slope of the straight line is
m = tanθ
Slope of a straight line when two points are given :
Let A(x_{1}, y_{1}) and B(x_{2}, y_{2}) be two distinct points on the line segment AB. Then, the slope of the line segment AB is
Slopes of parallel lines :
Let m_{1} and m_{2} be the slopes of two lines. If the two lines are parallel, then
m_{1} = m_{2}
That is, if two lines are parallel, then the slopes are equal.
Slopes of perpendicular lines :
Let m_{1} and m_{2} be the slopes of two lines. If the two lines are perpendicular, then
m_{1}m_{2} = -1
That is, if two lines are perpendicular, then the product of slopes equals to -1.
Equation of x-axis :
y = 0
Equation of y-axis :
x = 0
Equation of a straight line parallel to x-axis :
y = k
Equation of a straight line parallel to y-axis :
x = c
Equation of a line in general form :
Ax + By + C = 0
Equation of a line in standard form :
Ax + By = C
Equation of a line in slope-intercept form :
y = mx + b
where 'm' is the slope and 'b' is the y-intercept.
Equation of a line in point-slope form :
y - y_{1} = m(x - x_{1})
where 'm' is the slope and (x_{1}, y_{1}) is a point on the line.
Equation of a line in intercept form :
where 'a' is the x-intercept and 'b' is the y-intercept.
Angle between two straight lines :
Let the equations of the two straight lines be
y = m_{1}x + b_{1}
y = m_{2}x + b_{2}
Then the angle between these two straight lines is
Distance between a point and a line :
The distance from a point P(x_{1}, y_{1}) to a line ax + by + c = 0 is
Distance between two parallel lines :
Let the equations of two parallel lines be
ax + by + c_{1} = 0
ax + by + c_{2} = 0
Then the distance between these two parallel lines is
Equation of a circle in general form :
x^{2} + y^{2} + 2gx + 2fy + c = 0
where the center is C(-g, -f) and the radius is
Equation of a circle in standard form :
(x - h)^{2} + (y - k)^{2} = r^{2}
where the center is C(h, k) and the radius is r.
Equation of a circle in standard form with center (0, 0) :
x^{2} + y^{2} = r^{2}
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