Coordinate Geometry 2

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In Coordinate Geometry we define the coordinates of a point in a plane with reference to two mutual perpendicular lines in the same plane. It also includes plotting of points in the plane which is used to draw the graphs of linear equations in Cartesian plane. Coordinate Geometry

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Rectangular or Cartesian Co-ordinates of a Point The French Mathematician and philosopher Rene Descartes first published his book La Geometric in 1637 in which he use algebra in the study of geometry. This he did by representing points in a plain by ordered pairs of real numbers, called Cartesian coordinates Cartesian Co-ordinates of a Point

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Cartesians Coordinates Axes Let X’OX and Y’OY be two mutually perpendicular lines through a point O in the plane of a graph paper as shown in the fig. The lines X’OX is called the X’axis or axis of X the line Y’OY is known as the Y-axis or the axis of Y, and the two lines X’OX and Y’OY taken together are called the coordinate axes or the axes of coordinates. The point O is called origin . Quadrants T he coordinate axes X’OX and Y’OY divide the plane of the graph paper into four regions XOY, X’OY ,X’OY’ and Y’OX. These four regions are called the Quadrants. These regions XOY, X’OY ,X’OY’ and Y’OX are known as the first , second, third and fourth quadrants respectively .

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Cartesians Coordinates of a Point Let X’OX and Y’OY be the coordinate x=axes and let P be any point in the plane of the paper. Draw PM perpendicular to X’OX and PN perpendicular to Y’OY. The length of the line segment OM is called the X-coordinate or abscissa of point P is and the length of the directed line segment ON is called the Y-coordinate or ordinate of point P. If OM=3 Units and ON=5 Units ,then the X-coordinate or abscissa of point P is 3 and the Y-coordinate or ordinate of P is 5 and we say that the coordinate of P are (3,5).Note that (3,5) is an ordered pair in which the positions of 3 and 5 cannot be interchanged.

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Convention Of Signs Let X’OX and Y’OY be the coordinate axes. As discussed earlier that the regions XOY, X’OY’, Y’OX are known as the first , the second, the third and the fourth quadrants respectively. The ray OX is taken as positive X-axis, OX’ as the negative X-axis, OY as positive Y-axis and OY’ as negative Y-axis This means that any distance measured along OX will be taken as positive and the distance moved along OX’ will be negative. Similarly the distance moved along or parallel to OY will be taken as positive and the distance along OY’ will be negative.

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IN FIRST QUADRANT: X>0, Y>0 IN SECOND QUADRANT: X<0, Y>0 IN THIRD QUADRANT: X<0, Y<0 IN FOURTH QUADRANT: X>0, Y<0

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Plotting Of Points ALGORITHM STEP 1: Draw Two mutually perpendicular lines on the graph paper, one horizontal and the other vertical STEP 2 : Mark their intersection point as O (ORIGIN). The horizontal line as X’OX and the vertical line as Y’OY. The line X’OX is the X-axis and the line Y’OY as the Y-axis. STEP 3: Choose a suitable scale on X-axis and Y-axis and mark the points on both the axes

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STEP 4:Obtain coordinates of the point which is to be plotted. Let the point be (a, b). To plot this point, start from the origin and move ‘|a|’ units along OX or OX’ according as ‘a’ is positive or negative. Suppose we arrive at point M. From point M move vertically upward or downward through |b| units according as b is positive or negativeThe point where we arrive finally is the required point P (a, b)

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Q. Plot the following points on a graph paper: (3, 4) (-2, 3) (-5, -2) (4,-3)

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The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points . x 2 and y 2 are the x,y coordinates for one point x 1 and y 1 are the x,y coordinates for the second point d is the distance between the two points Distance Formula

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PQN is a right angled  .  PQ 2 = PN 2 + QN 2 PQ 2 = (x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 P(x 1 , y 1 ) Q(x 2 , y 2 ) x 1 X X’ Y’ O Y x 2 y 1 y 2 N y 2 -y 1 (x 2 -x 1 ) M R

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Given two end points of line segment A(x1, y1) and B (x2, y2) you can determine the coordinates of the point P(x, y) that divides the given line segment in the ratio m:n using Section Formula. The midpoint of a line segment divides it into two equal parts or in the ratio 1:1.The line joining the vertex to the midpoint of opposite side of a triangle is called Median.  Three medians can be drawn to a triangle and the point concurrency of medians of a triangle is called Centroid denoted with G.The centroid of a triangle divides the median in the ratio 2:1. Section Formula

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By AA Criterion  AHP ~ PKB A(x 1 , y 1 ) B(x 2 , y 2 ) X’ Y’ O Y P(x, y) m n : L N M H K X Internal Division

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By AA Criterion  PAH ~ PBK P divides AB externally in ratio m:n External Division X’ Y’ O Y A(x 1 , y 1 ) P(x, y) B(x 2 , y 2 ) L N M H K X

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Area of a Triangle X X’ Y’ O Y A(x 1 , y 1 ) C(x 3 , y 3 ) B(x 2 , y 2 ) M L N Area of  ABC = Area of trapezium ABML + Area of trapezium ALNC - Area of trapezium BMNC

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Area of  ABC =Area of trapezium ABML+ Area of trapezium ALNC- Area of trapezium BMNC Thus, the area of Δ ABC is the numerical value of the above expression. Presentation b y : Rishabh Puri X-C Roll.No.9

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RISHABH PURI X-C Roll.No.9 THANK YOU

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