1 Distance formula :
Distance between two point P(x1, y1) and Q(x2 , y2) is given by d(P,Q) = PQ
Note:
(i) d(P,Q) ³ 0
(ii) d(P,Q) =0 Û P = Q
(iii) d(P,Q) = d(Q,P)
(iv) Distance of a point (x,y) from origin
(0,0) = Ö(x2+y2)
2 Use of Distance formula :
(a) In Triangle : Calculate AB , BC , CA
(i) If AB = BC = CA , then Δ is equilateral
(ii) If any two sides are equal then Δ is isosceles
(iii) If sum of square of any two sides is equal to the third, then Δ is right triangle
(iv) Sum of any two equal to left third they do not form a triangle i.e. AB = BC + CA or BC = AC + AB or AC = AB + BC here point are collinear .
(b) In Parallelogroam :
Calculate Ab , BC , CD and AD.
(i) If AB = CD , AD = BC , then ABCD is a parallelogram.
(ii) If AB = CD , AD= BC and AC = BD , then ABCD is a rectangle.
(iii) If AB = BC = CD = AD , then ABCD is a rhombus.
(iv) If AB = BC = CD = AD and AC = BD , then ABCD is a square.
(c) For circumcentre of a triangle :
Circumcentre of a triangle is equidistant from vertices i.e. PA = PB = PC. Here P is circumcenter and PA is radius.
(i) Circumcenter of an acute angled triangle is inside the triangle .
(ii) Circumcentre of a right triangle is mid point of the hypotenuse .
(iii) Circumcentre of an obtuse angled triangle is outside the triangle .
3 Section formula :
(i) Internally :
AP/BP = m/n = l , Here l > 0
(ii) Externally :
AP/BP = m/n = l
(iii) Coordinates of mid point of PQ are 
(iv) the line ax + by + c =0 divides the line joining the point
(x1, y1) & (x2, y2) in the ratio = 
(v)For parallelogram – midpoint of diagonal AC = mid point of diagonal BD
(vi) Coordinates of centroid G 
(vii) Coordinates of incentre I 
(viii) Coordinates of orthocenter are obtained by solving the equation of any two altitudes.
4 Area of Triangle :
The area of trinagle ABC with vertices A(x1, y1) , B(x2, y2) and C(x3, y3) .
 (Determinant method)
 = (1/2 ) [ x1y2 + x2y3 + x3y1 - x2y1 - x1y3 ]
Note :
(i) Three point A, B, C are collinear if area of triangle is zero
(ii) If in a triangle point arrange in anticlockwise then values of Δ be + ve and if in clockwise then Δ will be – ve.
5 Area of Polygon :
Area of polygon having vertices (x1, y1) , (x2, y2), (x3, y3) …… (xn , yn) is given by area
 points must be in order.
6 Rotational Transformation :
If coordinates of any point P(x, y) with refernce to new axis will be (x’ , y’) then
| | x ¯ | y ¯ |
| x’ | ® | cos q | sin q |
| y’ | ® | -sin q | cos q |
7 Some important points :
(i) Three pts . A, B, C are collinear , if area of triangle is zero
(ii) Centroid G of ΔABC divides the median AD or BE or CF in the ratio 2: 1
(iii) In an equilateral triangle , orthocenter , centroid,circumcentre, incentre coincide .
(iv) Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining orthocenter and circumcentre in the ratio 2 :1
(v) area triangle formed by coordinate axes the line ax + by + c = 0 is (c2 /2ab) .
|
|
