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Analytical Geometry

Introduction

Analytical geometry, also called co-ordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra, and the Cartesian co-ordinate system. It is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.

Distance between Two Points

One of the simplest things that can be done with analytical geometry is to calculate the distance between two points. Distance is a number that describes how far apart two point are. For example, point PP has co-ordinates (2,1)(2,1) and point QQ has co-ordinates (-2,-2)(-2,-2). How far apart are points PP and QQ? In the figure, this means how long is the dashed line?

Figure 1
Figure 1 (MG10C14_015.png)

In the figure, it can be seen that the length of the line PRPR is 3 units and the length of the line QRQR is four units. However, the PQRPQR, has a right angle at RR. Therefore, the length of the side PQPQ can be obtained by using the Theorem of Pythagoras:

P Q 2 = P R 2 + Q R 2 P Q 2 = 3 2 + 4 2 P Q = 3 2 + 4 2 = 5 P Q 2 = P R 2 + Q R 2 P Q 2 = 3 2 + 4 2 P Q = 3 2 + 4 2 = 5
(1)

The length of PQPQ is the distance between the points PP and QQ.

In order to generalise the idea, assume AA is any point with co-ordinates (x1;y1)(x1;y1) and BB is any other point with co-ordinates (x2;y2)(x2;y2).

Figure 2
Figure 2 (MG10C14_016.png)

The formula for calculating the distance between two points is derived as follows. The distance between the points AA and BB is the length of the line ABAB. According to the Theorem of Pythagoras, the length of ABAB is given by:

A B = A C 2 + B C 2 A B = A C 2 + B C 2
(2)

However,

B C = y 2 - y 1 A C = x 2 - x 1 B C = y 2 - y 1 A C = x 2 - x 1
(3)

Therefore,

A B = A C 2 + B C 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 A B = A C 2 + B C 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2
(4)

Therefore, for any two points, (x1;y1)(x1;y1) and (x2;y2)(x2;y2), the formula is:

Distance=(x1-x2)2+(y1-y2)2(x1-x2)2+(y1-y2)2

Using the formula, distance between the points PP and QQ with co-ordinates (2;1) and (-2;-2) is then found as follows. Let the co-ordinates of point PP be (x1;y1)(x1;y1) and the co-ordinates of point QQ be (x2;y2)(x2;y2). Then the distance is:

Distance = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 2 - ( - 2 ) ) 2 + ( 1 - ( - 2 ) ) 2 = ( 2 + 2 ) 2 + ( 1 + 2 ) 2 = 16 + 9 = 25 = 5 Distance = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 2 - ( - 2 ) ) 2 + ( 1 - ( - 2 ) ) 2 = ( 2 + 2 ) 2 + ( 1 + 2 ) 2 = 16 + 9 = 25 = 5
(5)

The following video provides a summary of the distance formula.

Figure 3
Khan academy video on distance formula

Calculation of the Gradient of a Line

The gradient of a line describes how steep the line is. In the figure, line PTPT is the steepest. Line PSPS is less steep than PTPT but is steeper than PRPR, and line PRPR is steeper than PQPQ.

Figure 4
Figure 4 (MG10C14_017.png)

The gradient of a line is defined as the ratio of the vertical distance to the horizontal distance. This can be understood by looking at the line as the hypotenuse of a right-angled triangle. Then the gradient is the ratio of the length of the vertical side of the triangle to the horizontal side of the triangle. Consider a line between a point AA with co-ordinates (x1;y1)(x1;y1) and a point BB with co-ordinates (x2;y2)(x2;y2).

Figure 5
Figure 5 (MG10C14_018.png)

Gradient=y2-y1x2-x1=y2-y1x2-x1

We can use the gradient of a line to determine if two lines are parallel or perpendicular. If the lines are parallel (Figure 6a) then they will have the same gradient, i.e. mAB == mCD. If the lines are perpendicular (Figure 6b) than we have: -1mAB=mCD-1mAB=mCD

Figure 6
Figure 6 (geom.png)

For example the gradient of the line between the points PP and QQ, with co-ordinates (2;1) and (-2;-2) (Figure 1) is:

Gradient = y 2 - y 1 x 2 - x 1 = - 2 - 1 - 2 - 2 = - 3 - 4 = 3 4 Gradient = y 2 - y 1 x 2 - x 1 = - 2 - 1 - 2 - 2 = - 3 - 4 = 3 4
(6)

The following video provides a summary of the gradient of a line.

Figure 7
Gradient of a line

Midpoint of a Line

Sometimes, knowing the co-ordinates of the middle point or midpoint of a line is useful. For example, what is the midpoint of the line between point PP with co-ordinates (2;1)(2;1) and point QQ with co-ordinates (-2;-2)(-2;-2).

The co-ordinates of the midpoint of any line between any two points AA and BB with co-ordinates (x1;y1)(x1;y1) and (x2;y2)(x2;y2), is generally calculated as follows. Let the midpoint of ABAB be at point SS with co-ordinates (X;Y)(X;Y). The aim is to calculate XX and YY in terms of (x1;y1)(x1;y1) and (x2;y2)(x2;y2).

Figure 8
Figure 8 (MG10C14_019.png)
X = x 1 + x 2 2 Y = y 1 + y 2 2 S x 1 + x 2 2 ; y 1 + y 2 2 X = x 1 + x 2 2 Y = y 1 + y 2 2 S x 1 + x 2 2 ; y 1 + y 2 2
(7)

Then the co-ordinates of the midpoint (SS) of the line between point PP with co-ordinates (2;1)(2;1) and point QQ with co-ordinates (-2;-2)(-2;-2) is:

X = x 1 + x 2 2 = - 2 + 2 2 = 0 Y = y 1 + y 2 2 = - 2 + 1 2 = - 1 2 S is at ( 0 ; - 1 2 ) X = x 1 + x 2 2 = - 2 + 2 2 = 0 Y = y 1 + y 2 2 = - 2 + 1 2 = - 1 2 S is at ( 0 ; - 1 2 )
(8)

It can be confirmed that the distance from each end point to the midpoint is equal. The co-ordinate of the midpoint SS is (0;-0,5)(0;-0,5).

P S = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 0 - 2 ) 2 + ( - 0 . 5 - 1 ) 2 = ( - 2 ) 2 + ( - 1 . 5 ) 2 = 4 + 2 . 25 = 6 . 25 P S = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 0 - 2 ) 2 + ( - 0 . 5 - 1 ) 2 = ( - 2 ) 2 + ( - 1 . 5 ) 2 = 4 + 2 . 25 = 6 . 25
(9)

and

Q S = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 0 - ( - 2 ) ) 2 + ( - 0 . 5 - ( - 2 ) ) 2 = ( 0 + 2 ) ) 2 + ( - 0 . 5 + 2 ) ) 2 = ( 2 ) ) 2 + ( - 1 . 5 ) ) 2 = 4 + 2 . 25 = 6 . 25 Q S = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 0 - ( - 2 ) ) 2 + ( - 0 . 5 - ( - 2 ) ) 2 = ( 0 + 2 ) ) 2 + ( - 0 . 5 + 2 ) ) 2 = ( 2 ) ) 2 + ( - 1 . 5 ) ) 2 = 4 + 2 . 25 = 6 . 25
(10)

It can be seen that PS=QSPS=QS as expected.

Figure 9
Figure 9 (MG10C14_020.png)

The following video provides a summary of the midpoint of a line.

Figure 10
Khan academy video on midpoint of a line

Co-ordinate Geometry

  1. In the diagram given the vertices of a quadrilateral are F(2;0), G(1;5), H(3;7) and I(7;2).
    Figure 11
    Figure 11 (MG10C14_021.png)
    1. What are the lengths of the opposite sides of FGHI?
    2. Are the opposite sides of FGHI parallel?
    3. Do the diagonals of FGHI bisect each other?
    4. Can you state what type of quadrilateral FGHI is? Give reasons for your answer.
    Click here for the solution
  2. A quadrialteral ABCD with vertices A(3;2), B(1;7), C(4;5) and D(1;3) is given.
    1. Draw the quadrilateral.
    2. Find the lengths of the sides of the quadrilateral.
    Click here for the solution
  3. ABCD is a quadrilateral with verticies A(0;3), B(4;3), C(5;-1) and D(-1;-1).
    1. Show that:
      1. AD = BC
      2. AB DC
    2. What name would you give to ABCD?
    3. Show that the diagonals AC and BD do not bisect each other.
    Click here for the solution
  4. P, Q, R and S are the points (-2;0), (2;3), (5;3), (-3;-3) respectively.
    1. Show that:
      1. SR = 2PQ
      2. SR PQ
    2. Calculate:
      1. PS
      2. QR
    3. What kind of a quadrilateral is PQRS? Give reasons for your answers.
  5. EFGH is a parallelogram with verticies E(-1;2), F(-2;-1) and G(2;0). Find the co-ordinates of H by using the fact that the diagonals of a parallelogram bisect each other.
    Click here for the solution

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