9th 12345678900yuCoordinate Geometry.pptx

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1. For Grade-IX COORDINATE GEOMETRY Chapter-3 By Mr. Akash Yadav M.Sc , B.Ed TGT(MATHS) PM SHRI KV AFS BIDAR Contact - 8829953881 Email – [email protected]

2. CONTENTS: 1 . Introduction to Coordinate Geometry 2. Cartesian System 3. Plotting a Point in the Plane if its Coordinate are given 4.Summary

3. 1. Introduction to Coordinate Geometry: What is a Geometry? The Branch of Mathematics Concerned with the properties of and relationship between points, lines, planes, and figures What is coordinate? A pair of numbers that describe the position of a point on a coordinate plane by using the horizontal and vertical distances from the two reference axes. Usually represented by ( x,y ) the x-value and y-value. And O is the centre i.e (0,0) called origin. What is Coordinate Geometry? A coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates. The father of coordinate geometry was René Descartes

4. 2.Cartesian System/Cartesian Coordinate System : What is Cartesian coordinate system? A Cartesian system is the one which specifies a point plotted uniquely on a plane by a pair of numerical coordinates, having a pre-defined distance from the two fixed perpendicular coordinates, measured in the same unit of length.

5. Cartesian Plane: What is a Plane? Flat surface that extends infinitely along length and width. WIDTH Length Cartesian is two dimensional grid formed by drawing two number lines: One is perpendicular to each other. i . e X- By horizontal and Y- By Vertical and two point intersect at O (0,0) Then What is Cartesian Plane?

6. Abscissa and ordinate: In the coordinate system, the term “Ordinate” refers to the y -coordinate in  (x, y). In short, the distance along the y-axis is called the ordinate. There is a difference in two terms called coordinate and ordinate. The term “coordinate” represents the position of the point along the line, arc and so on, whereas the term “ordinate” represents the value of a coordinate on the y-axis The  ordinate  is the second component of an ordered pair. For example, if (x, y) is an ordered pair, then y is the ordinate here. An ordered pair is used to denote a point in the Cartesian plane. It is the second of the two terms (x, y) by which a point is defined, in a system of fixed  Cartesian Coordinate plane . The first coordinate (x), in the plane, is called the  abscissa . The Cartesian system designates each point in the XY plane, uniquely, by a pair of numerical coordinates. These points are at the distances from the two fixed axes, measured in the same unit of length. A Is the Coordinate

7. What Is a Cartesian Coordinate Plane? A Cartesian coordinate plane is a system of two perpendicular number  lines  on which we locate points. The Cartesian coordinate plane consists of two perpendicular lines, x-axis and y-axis. The horizontal line is called the  x-axis . The vertical line is called the  y-axis . The intersection  point  of these two axes is called origin and is represented by  O . The axes split through the origin, which is marked as 0 on the graph. The side on the right of 0 is positive, while everything on the left is negative. Similarly, as you go higher, the numbers are positive, while the negative numbers are written below 0 The positive x-axis has positive numbers such as 1, 2, 3, 4,….. The negative x-axis has negative numbers such as -1, -2, -3, -4,….. The positive y-axis has positive numbers such as 1, 2, 3, 4,….. The negative y-axis has positive numbers such as -1, -2, -3, -4,….. A Cartesian coordinate plane can simply be called a coordinate plane or a plane. It represents a 2-dimensional (2D) plane. The Cartesian coordinate system is sometimes also referred as the Cartesian coordinate graph.

8. What is a Quadrants ? Quadrant is the region enclosed by the intersection of the X-axis and the Y-axis. On the cartesian plane when the two axes, X-axis and Y-axis, intersect with each other at 90º there are four regions formed around it, and those regions are called quadrants. So, every plane has four quadrants each bounded by half of the axes. Each quadrant is denoted by Roman numerals and named as Quadrant I, Quadrant II, Quadrant III, and Quadrant IV based on their position with respect to the axes Definition: A quadrant can be defined as a region/part of a cartesian plane which is obtained when the two axes intersect each other. It is used to determine the position of a point in a plane.

9. 3. Plotting a Point in the Plane if its Coordinate are Given A point in a plane contains two components where order matters! It comes in the form ( x,y ) where x comes first, and y comes second. The x-value tells how the point moves either to the right or left along the x-axis .  This axis is the  main horizontal   line  of the rectangular axis or Cartesian plane. The y-value tells how the point moves either up or down along the y-axis. This axis is the  main vertical line  of the rectangular axis or Cartesian plane. The numbers on a  coordinate  grid are used to locate  points . Each  point  can be identified by an ordered pair of numbers; that is, a number on the x-axis called an x- coordinate , and a number on the y-axis called a y- coordinate . Ordered pairs are written in parentheses (x- coordinate , y- coordinate )

10. How do you represent the points on the XY-Plane? ( x,y ) For Example -1 Write the coordinates of each of the points P,Q,R,S,T and O from the figure? The coordinates of the points P,Q,R,S,T and O P=(1,1) Q=(-3,0) R=(-2,-3) S=(2,1) T=(4,-2) O=(0,0)

11. 2. Points A(5,3) , B(-2,3) and D(5,-4) are the three vertices of a rectangle ABCD. Plot these points on a graph paper and hence find the coordinates of the vertex C.

12. EXERCISE 3.1 1. How will you describe the position of a table lamp on your study table to another person? Solution: y x (25,30) Lamp on Table Fist Draw a x and y line and mark the points and measure the lamp distance from x and y axis the distance of the lamp from y-axis is 30 cm and x-axis 25 cm

13. Solution: W

14. Example1: William wants to throw a ball into a bucket which is placed on the coordinates (3, 10). If William is standing on the coordinates (3, 5), how far will he have to throw the ball? Solution To calculate how far William will have to throw the ball, we need to calculate the distance between William and the bucket. The x-coordinate of William's location and the bucket is 3 Thus, they are on a line parallel to the y-axis. Hence, the distance between William and the bucket can be calculated as the difference of the y-coordinates of both the points. 10−5=5units Thus, Willam has to throw the Ball 5 unit way

15. ∴   a =3 Solution We know that both the points P and Q lie on a line parallel to the x-axis, thus, their y-coordinates are the same. 7− a =4 a =7−4 a =3 Example2 : If point P (3,4) and point Q (5,7−a) lie on a line parallel to the x-axis, then what is the value of a? Example3 : Jacob and Ethan want to make a frame using the coordinates (1,2),(3,2),(3,0),(1,0).Based on the coordinates, Jacob says that the frame will be a square while Ethan says that the frame will be a parallelogram. Can you identify who is right? Solution We need to draw the above coordinates on a Cartesian plane to check the shape they will form. We can clearly see that the figure thus obtained is a  square as all the four sides are equal and all the four interior angles are 90∘ There fore, Jocob is right.

16. EXAMPLE 1 Solution: Scale: 1 unit along X-axis=1CM 1 unit along Y-axis =1CM

17. EXAMPLE 2 Solution:

18. EXAMPLE 3 Solution:

19. 4 Summary:

20. References: NCERT Text book. Google sources. THANK YOU