10th Class Mathematics Coordinate Geometry Co-ordinate Geometry

Co-ordinate Geometry

Category : 10th Class

Co-ordinate Geometry

 

 

 

  • The branch of geometry that sets up a definite correspondence between the position of a point in a plane and a pair of algebraic numbers called coordinates is called Coordinate geometry.
  • The distance of a point from the Y-axis is called the X-coordinate or Abscissa.
  • The distance of a point from the X-axis is called the Y-coordinate or ordinate.
  • The coordinates of a point on the X-axis are of the form (x, 0) and of a point on the Y-axis are of the form (0, y).
  • The abscissa and ordinate of a point taken together is known as coordinates of a point.
  • The point of intersection of the axes of coordinates is called the origin.
  • The quarter plane that results from the division of the plane by the coordinate axes is called a quadrant.
  • Signs of x and y coordinates in the four quadrants:

Quadrant

X-coordinate

y-coordinate

Sign of the coordinates of the point

I

Positive

Positive

\[(+,+)\]

II

Negative

Positive

\[(-,+)\]

III

Negative

Negative

\[(-,-)\]

IV

Positive

Negative

\[(+,-)\]

 

Distance Formula:

  • The distance between two points \[A({{x}_{1}},{{y}_{1}})\] and \[B({{y}_{2}},{{y}_{2}})\] is given by

\[AB=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}\]

  • The distance of the point \[p(x,y)\]from the origin 0 (0,0) is given by \[OP=\sqrt{{{x}^{2}}+{{y}^{2}}}\].
  • Properties of various types of quadrilaterals: a quadrilateral is a

(a) Rectangle: If its opposite sides are equal and the diagonals are equal.

(b) Square: If all its sides are equal and the diagonals are equal.

(c) Parallelogram: If its opposite sides are equal.

(d) Parallelogram but not a rectangle: If its opposite sides are equal and the diagonals are not equal.

(e) Rhombus but not a square: If all its sides are equal and the diagonals are not equal.

  • Collinear points: Points are said to be collinear if they lie on the same straight line.
  • Test for co linearity of given points: Given three points A, B and C, find the distances AB, BC and AC. If the sum of any two of these distances is equal to the third distance, then the given points are collinear.
  • Section formula: 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 \[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\]and \[y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\]

Note: The coordinates of a point \[p(x,y)\]which divides the line segment joining \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]externally in the ratio m: n are given by \[x=\frac{m{{x}_{2}}-n{{x}_{1}}}{m+n}\]and \[y=\frac{m{{y}_{2}}-n{{y}_{1}}}{m-n}\].

  • Midpoint formula: The coordinates of the midpoint M of a line segment AB joining the points \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]are \[M\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]
  • Centroid: The point of intersection of the medians of a triangle is called its centroid. The centroid of a triangle divides each median in the ratio 2:1.
  • The coordinates of the centroid of a triangle are \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]
  • Area of a triangle: If \[A({{x}_{1}},{{y}_{1}}),B({{x}_{2}},{{y}_{2}})\]and \[C({{x}_{3}},{{y}_{3}})\]are the vertices of a triangle, its area is given by\[\frac{1}{2}|{{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})|\]
  • If the area of triangle is 0, then the three points that form its vertices are collinear. In other words, three points are collinear if \[{{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})=0\].
  • Incentre: The points of concurrence of the bisectors of the angles of a triangle are called the incentre of the triangle. The coordinates of the incentre of a triangle are \[\left( \frac{a{{x}_{1}}+b{{x}_{2}}+c{{x}_{2}}}{a+b+c},\frac{a{{y}_{1}}+b{{y}_{2}}+c{{y}_{3}}}{a+b+c} \right)\]where a, b and c are the sides of\[\Delta \text{ABC}\].

 

Notes - Co-ordinate Geometry
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