IMPORTANT TERMS, DEFINITIONS AND RESULTS

  1. In the rectangular coordinate system, two number lines are drawn at right angles to each other. The point of intersection of these two number lines is called the origin whose coordinates are taken as (0, 0). The horizontal number line is known as the x-axis and the vertical one as the y-axis.
  2. In the ordered pair (p, q), p is called the x-coordinate or abscissa and q is known as y-coordinate or ordinate of the point.
  3. The coordinate plane is divided into four quadrants.

4. The abscissa of a point is its perpendicular distance from y-axis.

5. The ordinate of a point is its perpendicular distance from x-axis.

6. The abscissa of every point situated on the right side of y-axis is positive and the abscissa of every point situated on the left side of y-axis is negative.

7. The ordinate of every point situated above x-axis is positive and that of every point below x-axis is negative.

8. The abscissa of every point on y-axis is zero.

9. The ordinate of every point on x-axis is zero.

10. The distance between any two points P(x_1,y_1) and Q(x_2,y_2) is given by
PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
or PQ=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\Rightarrow PQ\ \sqrt{\text{(Difference of absissae)}^2+\text{(Difference of ordinats)}^2}

11. If O(0, 0) is the origin and P(x, y) is any point then from the above formula, we have :
OP=\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2}

12. In order to prove that a given figure is a :
(i) square, prove that four sides are equal and the diagonals are equal.
(ii) rhombus, prove that the four sides are equal.
(iii) rectangle, prove the opposite sides are equal and the diagonals are also equal.
(iv) parallelogram, prove that the opposite sides are equal
(v) parallelogram but not a rectangle, prove that its opposite sides are equal but diagonals are not equal.

13. Three points A, B and C are said to be collinear, if they lie on the same straight line.

14. For three points to be collinear, the sum of the distances between two pairs of points is equal to the third pair of points.

15. Three points will make :
(i) a scalene triangle, if no two sides of the triangle are equal.
(ii) an isosceles triangle, if any two sides are equal
(iii) an equilateral triangle, if all the three sides are equal.
(iv) a right triangle, if sum of the squares of any two sides is equal to the square of the third side.

16. The coordinates of the point P(x, y) which divides the line segment joining A(x_1, y_1) and B(x_2, y_2) internally in the ratio m : n, are given by :

x=\dfrac{mx_2+nx_1}{m+n},\ y=\dfrac{my_2+ny_1}{m+n}

17. The coordinates of the mid-point M of a line segment AB with end points A(x_1, y_1) and B(x_2, y_2) are :

\left[\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right]

18. The point of intersection of the medians of a triangle is called its centroid.

19. The coordinates of the centroid of the triangle whose vertices are (x_1, y_1), (x_2, y_2) and (x_3, y_3) are given by
\left[\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\right]

20. The area of \Delta ABC with vertices A(x_1,y_1),\ B(x_2,y_2) and C(x_3,y_3) is given by : area (\triangle ABC)=\left|\dfrac{1}{2}\{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\}\right|

Since area of a triangle cannot be negative we consider the absolute or numerical value of the area.

21. Three given points A(x_1, y_1), B(x_2, y_2) and C(x_3, y_3), are collinear if

\Rightarrow  area of \Delta ABC=0
\Rightarrow\dfrac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]=0

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