Coordinate Geometery

Published Date: 02 Nov 2017

Coordinate geometry is used to analyse geometric shapes, For example describing the position of points on a 2-dimensional surface (piece of paper). Position of points can be referred to in one of these three ways: fixed point: (4,8), generic fixed points (x2,y2) (x3,y3, general points (x,y), this means that the point can be anywhere along a shape distance between 2 points.

The distance formula is used to find the distance between any two points on a graph. This comes from applying the Pythagorean Theorem to a right triangle ABC to find the distance between A and B. We want to find the distance between A (x2,y2) and B(x3,y3). However if you look at the diagram above we have added an extra point C, so that triangle ABC is a right angle triangle. Using the Pythagorean theorem we can now find the length of the line A,B in the terms of A,B and C.

AB2=AC2+BL2

Remember that the location of A (x2,y2) and the location of B(x3,y3). Therefore the distance between B and C is x3-x2 and the distance between B and C is y3-y2. Therefore, the above equation can be re-written:

Ab2=(x3-x2)2 + (y3-y2)2

From looking at this equation, we can tell that the distance between A and B can be expressed by: ab=x3-x2-(y3-y3)

Below is an example:

Distance between (3,6) and (-3,7) is: ab=-3-32+7-62 = -52+12 = 25+2 =27 =5.10 (2 d.p)

Midpoint

This is used to find the midpoint between any two points. To find the midpoint of the X coordinate, you must take the average of the X coordinate of the points A and B (this is half way between x2 and x3, from the diagram above). To find the midpoint of the Y coordinate you must take the average of the Y coordinate of the points A and B (This is half way between Y2 and Y3, from the diagram above).

Therefore we can rewrite the following as equations:

x2+x32

The X coordinates of M y2+y32, the y coordinates of M From the following equations you can find the midpoint of any line.

Below is an example:

Find the midpoint of (2,6) and (-3,9) 2+-32=-0.5 6+92=7.5 Midpoint = (0.5,7.5)

Gradient

Gradient is used in a straight line to measure its slops in relations to the x-axis. The gradient line can be either positive or negative. An uphill slope shows a positive gradient. This means the x and y axis has had a positive increase. Whereas a downhill slope shows a negative gradient, this means the x-axis has a positive increase whereas the y-axis has a negative decrease.

How to calculate the gradient of a line:

To work out the gradient you will need to divide the increase of the Y coordinate by the increase of the x coordinate. Equation for working out the gradient of a line: Y3-y2X3-X2.

If line 1 and 2 are parallel to each other then they will have the same angle of indication to the x-axis. Therefore they have equal gradients. Equation of a straight line Y=mx+c M= gradient of the line c= line intercepts the y-axis. If we are given the value for m and c, we are able to write the equation for a line e.g. Y=3x + 4.

Intersection of two lines

Intersection is the point where two lines cross. Therefore if the point of intersection is at (x,y), meaning x and y are the only points on the line where x will give the same value of y in the equation for both lines. At the point of intersection the equations of the two lines are equal. Hence, m1x + c1 = m2x + c2.

For example:

Find where the following two lines meet:

Line 1: y = 2x + 5 Line 2: y = -0.5x + 10 Using m1x + c1 = m2x +c2 we know that: 2x + 5 (equation of line 1) = -0.5x + 10 (equation of line 2) This solves to give: x = 2. So we know the lines intersect where x = 2. Now all we need is the y value. To do this insert x = 2 into the equation for either line. If we use line 1 we get, y = 2 x 2 + 5 = 9 We now know that at the point of intersection, x = 2 and y = 9, or (2,9).

Curves

The following diagrams show the minimum and maximum values of y in the region of the curves. These points may be calculated, as they are the points where the differential of the equation of the curve equals zero. The other point you will be able to identify is the point of inflection. At this point the differential of the equation of the curve does equal zero, but in the case of a point of inflection the gradient does not change from positive to negative, but carries on as a positive or negative gradient. A cubic curve can have various positions in relation to the x-axis.