• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12
  13. 13
  14. 14
  15. 15
  16. 16
  17. 17

Circles Portfolio. The purpose of this assignment is to investigate several positions of points in three intersecting circles in order to discover a general statement

Extracts from this document...


Mathematics SL, Portfolio Type 1Circles


The purpose of this assignment is to investigate several positions of points in three intersecting circles in order to discover a general statement. There are three circles: circle C1 has center O and radius r, circle C2 has center P and radius OP, and circle C3 has center A and radius r (the same radio as C3). Point P' is the intersection of C3 with (OP).

All diagrams in this portfolio were created with Geogebra. To give a clear display of the length of each segment the diagrams are presented in the Cartesian coordinate system using point O as origin and (OP) as x-axis.

At first I will try to find a general statement representing the relation of OP’, OP and r in three examples, in which r will remain constant and OP will have three different values.


Diagram  1: r=1, OP=2

The method I will use to find OP’ will include the cosine function used in two triangles I will sketch in the first three diagrams:

...read more.













Table 1: Results from first part of investigation

In the first part of the investigation with the help of the cosine function we find out that in all cases


 are similar triangles, as in each case they share the same angle measurements (AOP=OAP’, OPA =OAP’ and OAP =AP’O) and the corresponding sides are in proportion (image05.png


When observing the results we find that if each number of OP is multiplied by their correspondent OP’ the result is 1, so this pattern can be generalized by:      image01.png


In the following part of the investigation, I will change the roles of r and OP so that OP will remain constant and r will have three different values.

All values of OP’ were found with Geogebra.


  P (0, 2)



   is an equilateral triangle, as      OP=P’A=AO=2


        P’ (0, 4.5)



       P’ (0, 8)

















Table 2: Results from second part of investigation

Verifying if the statement found in the previous part of the investigation is consistent:

Diagram 4:                                Diagram 5:                                 Diagram 6:


=OP’                                 image01.png

=OP’                           image01.png


...read more.


-Example 9: Radius r is a negative number.

-Example 10:  


                                                      Diagram  11: r=9, OP=4.49

Diagram 11 shows that point P’ is undefined. The combination of values of r=9 and OP=4.5 causes the two circles with radius r to form one single circle. This makes it impossible to define the intersection of C3 with (OP), as C3 does not exist.

The general statement OP’=image01.png

is valid only if r

If r

  • (diagram 11), the value of OP’ is undefined.
  • If r and OP are not bigger than 0 the value of OP’ is undefined.

If r = 2 · OP (diagram 7), the center of C3 is situated on the x-axis. 2·OP is the biggest value r can reach. Triangles


  •  cannot be sketched, but the general statement can still be used for finding OP’.

If r

  •  (diagrams 1, 2, 3, 4, 5, 6, 8, 9,10), OP’ can be found through the general formula.

The process of finding the general statement starts already in the first part of the investigation. With help of the cosine function we find out that in all cases


are similar triangles. The demonstration of the corresponding sides being in proportion can be represented by(image05.png

). By simplifying image06.png

we get to the general statement OP’=image01.png


...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. This portfolio is an investigation into how the median Body Mass Index of a ...

    girls in Shaanxi, after slight changes to values of vertical displacement and horizontal phase shift. The majority of the people in USA live in urban conditions, so it is understandable that given similar living conditions the data for urban girls in Shaanxi follows the same pattern.

  2. Math SL Circle Portfolio. The aim of this task is to investigate positions ...

    By using the distance formula, , we can use a algebraic approach to system of equations to solve for the unknown coordinates. Through this, we can find out the coordinates of A. = 1, A(a, b), O(0, 0) Distance formula: 1= 1= 1= + = 2, A(a, b), P(2, 0)

  1. Maths Investigation: Pascals Triangles

    1 + 2x + x^2 Row 3:(x+1)^3 = 1 + 3x + 3x^2 + x^3 Row 4:(x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 Row 5:(x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 ...

  2. This essay will examine theoretical and experimental probability in relation to the Korean card ...

    There are two different situations of getting these hands, getting January with? first and getting either of March or August with?. P(a) = 2 x P(player 1 getting January with ?) x P(player 2 getting any cards excluding March or August with?)

  1. Math IA Type 1 Circles. The aim of this task is to investigate ...

    For angle measures, exact values will be kept throughout the calculations. Utilizing exact values for angles will be beneficial because angle measures are used throughout the calculations but do not appear in the final answers; in this way the final answer will be extremely accurate because unnecessary rounding will be avoided.

  2. The purpose of this investigation is to create and model a dice-based casino game ...

    between one and p; therefore there are different ways to do this for each p out of total possible outcomes; therefore the probability that p is the highest number that player A rolls is . Similarly, in the probability that an integer q between one and six is recognized as

  1. This assignments purpose is to investigate how translation and enlargement of data affects statistical ...

    standard deviation will be affected because it has a greater effect on the variance. If the constant is smaller than 0, the mean and standard deviation would be a negative number. 1. The table below is a transformation of the values in the information given into a cumulative frequency table.

  2. In this portfolio, I am required to investigate the number of regions obtained by ...

    Therefore, I plotted the figures in the data table in the XY data set of geometry software Autograph to find the equation of the line.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work