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# Aptitude - Co-ordinate Geometry

## Position of a point in a plane

In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales - one running across the plane called the "x axis" and another right angle to it called the y axis. (These can be thought of as similar to the column and row in the paragraph above.) The point where the axes cross is called the origin and is where both x and y are zero.

On the x-axis, values to the right are positive and those to the left are negative. On the y-axis, values above the origin are positive and those below are negative. A point's location on the plane is given by two numbers; the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15. These are the coordinates of the point A, sometimes referred to as its "rectangular coordinates".

Note that the order is important; the x coordinate is always the first one of the pair.

## Distance between two points

If A(x_{1},y_{1}) and B (x_{2},y_{2}) be two points, then

AB =√(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2}

## Distance of a point from the origin

The distance of a points A(x, y) from the origin O(0, 0) is given by

OA =√(x^{2}+y^{2})

## Area of a triangle

If A(x_{1},y_{1}), B(x_{2},y_{2}) and C= (X_{3}, Y_{3})be three vertices of a ∆ABC, then its area is given by:

∆ = 1/2 {x_{1}(y_{2}- Y_{3})+ x_{2}(Y_{3}- Y_{1}) +X_{3}(y_{1}-y_{2})}

## Condition of co linearity of three points

Three points A(x_{1},y_{1}), B(x_{2},y_{2}) and C= (X_{3}, Y_{3}) are collinear if and only if ar(√ABC)= 0.

∴ A, B, C are collinear ⇒ x_{1}(y_{2}- Y_{3})+ x_{2}(Y_{3}- Y_{1}) +X_{3}(y_{1}-y_{2}) = 0

## Division of a line segment by a point

If a point p(x,y) divides the join of A(x_{1},y_{1}) and B(x_{2},y_{2}) in the ratio m:n, then

X= (mx_{2}+nx_{1})/m+n and Y =(my_{2}+ny_{1})/m+n

If A(x_{1},y_{1}) and B(x_{2},y_{2}) be the end points of a line segment AB, then the co-ordinates of midpoint of AB are

[(x_{1}+ x_{2})/ 2 , (y_{1}+ y_{2})/ 2]

## Centroid of a triangle

The point of intersection of all the medians of a triangle is called its centroid. If A(x_{1},y_{1}), B(x_{2},y_{2}) and C= (X_{3}, Y_{3}) be the vertices of ABC, then the co-ordinates of its centroid are { (1/3 (x_{1}+x_{2}+x_{3}),1/3 (y_{1}+y_{2}+Y_{3})}

## Various types of Quadrilaterals

A quadrilateral is

A rectangle if its opposite sides is equal and diagonals are equal.

A parallelogram but not a rectangle, if it's opposite sides are equal and the diagonals are not equal.

A square, if all sides are equal and diagonal are equal.

A rhombus but not a square, if all sides are equal and diagonals are not equal.

## Equations of lines

The equation of x-axis is y =0.

The equation of y –axis is x = 0.

The equation of a line parallel to y-axis at a distance a from it, is x= a.

The equation of a line parallel to x-axis at a distance b from it, is y= b.

The equation of a line passing through the points A(x

_{1},y_{1}) and B(x_{2},y_{2}) is y-y_{1}/ x-x_{1}= y_{2}-y_{1}/x_{2}-x_{1}. Slop of such a line is y_{2}-y_{1}/x_{2}-x_{1}.The equation of a line in slop intercept form is Y= mx+ c, where m is its slope.