* The distance between two points P(x1, y1) and Q(x2, y2) in the plane is denoted by PQ and is given by
Let A(x1, y1), B(x2, y2) be any two points in the plane, the coordinates of the point which divides in the ratio m : n are
* The division is called Internal if is positive and External if is negative.
* The area of where A(x1, y1), B(x2, y2), C(x3, y3).
* The area of quadrilateral
Concurrent lines, point of concurrency
Three or more straight lines are said to be concurrent if all the straight lines have exactly one point in common and this common point is called point of concurrency.
Median and Centroid
In a triangle the line segment joining a vertex to the mid point of the opposite side is called Median and the point of concurrency of Medians is called Centroid.
If A(x1, y1), B(x2, y2), C(x3, y3) are the vertices of triangle, then the centroid of ∆ ABC
Angular bisectors - Incentre - Excentre
The bisectors of the internal angles of a triangle ABC are concurrent. The point of concurrency is called Incentre, it is denoted by 'I'. The bisector of internal angle 'A' and the bisectors of external angles B and C are concurrent and the point of concurrency is called the Excenter opposite to the vertex 'A' and it is denoted by I1.
Let AB = c, BC = a, CA = b, then
Orthocenter and Altitudes
In a triangle the perpendicular from a vertex to the opposite side is called Altitude and point of concurrency is called Orthocentre.
Perpendicular bisector of sides - Circumcenter
The perpendicular bisector of sides are concurrent and the point of concurrency is called Circumcenter.
Let P(x, y) be a point in the original co-ordinate system.
If the axes are rotated through an angle '', then
i) The coordinates (x, y) of a point P are transformed as (x', y') = (x cos + y sin − xsin + y cos )
ii) The equation f(x, y) = 0 of the curve is transformed as
f (x' cos − y' sin , x' sin + y'cos ) = 0
* If the origin is shifted to (h, k) and then the axes are rotated through an angle '', then
i) The coordinates (x, y) of a point P are transformed as (x', y') = (x cos + y sin - h,
− x sin + y cos - k).
ii) The equation f (x, y) = 0 of the curve is transformed as f (x' cos - y' sin + h, -x sin + y cos - k) = 0.
* If we shift the origin to the point then the 1st degree terms of equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is removed.
* If the axes are rotated through an angle then the xy term is removed in the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
* Locus is the set of points that satisfy the given consistent geometric condition(s).
i.e., (i) Every point satisfying the given condition(s) is a point on the locus.
(ii) Every point on the locus satisfies the given condition(s).
* Equation of the locus of a point is an "algebraic equation" in x and y satisfied by the points (x, y) on the locus alone (and by no other point).
Translation of axes - Definition
The transformation obtained by shifting the origin to a given different point in the plane without changing the directions of coordinate axes there in is called as translation of axes.
Let P(x, y) be a point in original coordinate system. If we shift the origin to the point O'(h, k), then new coordinates of the point P are (x', y').
(i) x = x' + h, y = y' + k
(ii) The equation f(x, y) of the curve is transformed f(x' + h, y' + k) = 0
* Rotation of axes
The transformation obtained by rotating both the coordinate axes in the plane by an equal angle without changing the position of the origin is called rotation of axes.