To Find The Distance between Two Points in a Plane This formula is known as the Distance Formula  The distance between two points(x 1, y 1 ) and (x 2, y 2 ) = √(x 2 -x 1 ) 2 + (y 2 – y 1 ) 2 = √ (difference of x-coordinates) 2 + (difference of y-coordinates) 2 Let A ( x 1, y 1 ) and B (x 2, y 2 ) be two points in the plane Let AB = d. By definition of coordinates, OP = x 1, AP = y 1, OQ = x 2, BQ = y 2. From geometry, AR = PQ = OQ – OP = x 2 - x 1, and BR = BQ – RQ = BQ – AP = y 2 - y 1. In the right-angled ARB, AB 2 = AR 2 + BR 2  d 2 = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2. Distance Formula A B QP R O Y X (x 2, y 2 ) (x 1, y 1 ) y1y1 x1x1 y2y2 x2x2 d

Figure sides diagonals Square all 4 sides are equal are equal Rectangle opposite sides are equal Rhombus all 4 sides are equal are unequal Some important facts…

AP B mn Let AB be a line segment. Let P be a point on the line segment such that AP : PB = m : n Then, we can say that P divides AB in the ratio m : n (internally) Section Formula Remember, if AP : PB = m : n then AP / PB = m/n. So, any section by P can be expressed by AP:PB = m : n Section of a Line Segment

A B ML R O Y X Q N P (x 1, y 1 ) m n (x 2, y 2 ) (x, y) By applying the intercept theorem we get LM = AP or x-x 1 = m MN PB x 2 -x n n(x-x 1 ) = m(x 2 -x) nx – nx 1 = mx 2 – mx nx + mx = mx 2 + nx 1 x ( n+m) = mx 2 + nx 1 x= mx 2 + nx 1 Similarly, y = my 2 + ny 1 m + n m+n

A B ML R O Y X Q N P (x 1, y 1 ) m n (x 2, y 2 ) (x, y) APB mn (x 1,y 1 )(x, y)(x 2,y 2 ) The coordinates of the point P(x,y) which divides the line segment A (x 1,y 1 ) and B (x 2,y 2 ) in the ratio m:n are: Px= mx 2 + nx 1, Py = my 2 + ny 1 m+n m + n