Coordinate plane and coordinate of a point: The plane containing two mutually perpendicular lines X’OX and Y’OY intersecting each other at O is called coordinate plane. X’OX and Y’OY are called x-axis and y-axis respectively.
Test for geometrical figures:
(a) Isosceles triangle ⎯Prove that at least two sides are equal.
(b) Equilateral triangle ⎯ Prove that three sides are equal.
(c) Right triangle ⎯Prove that sum of squares of two sides is equal to square of third side.
(d) Square ⎯ Prove all sides are equal and diagonals are also equal.
(e) Rhombus ⎯ Prove all sides are equal and diagonals are unequal.
(f) Rectangle ⎯ Prove that the opposite sides are equal and diagonals are also equal.
(g) Parallelogram⎯ Prove that opposite sides are equal and diagonals are not equal.
Aid to memory: The above formula can be recollected easily by drawing an array as given alongside. Here we write coordinates of the vertices of ΔABC in two columns, repeating the coordinates of the first vertex.
Condition for collinearity of three points:
(a) Distance of formula based: Points A, B, C are collinear
AB + BC = AC i.e
., Sum of any two distance = third distance
(b) Area of triangle based: Points A, B, C are collinear
If ar (ΔABC) = 0
Some additional tips:
(a) Points of trisection: If a line segment is divided into three equal parts by two points, the points are said to be points of trisection.Clearly, if P and Q are points of trisection then