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

Math IA Type 1 Circles. The aim of this task is to investigate the positions of points in intersecting circles.

Extracts from this document...

Introduction

Math IA (SL Type I)CirclesMariya Lupandina

Aim: The aim of this task is to investigate the positions of points in intersecting circles.

Introduction

        This investigation will examine the length of OP’ as it changes with different values of r and OP. A general statement will be deciphered to represent this relationship, using trigonometry, and the validity of this statement will be tested. In the first part of the investigation, r will be held constant, OP will vary and a general statement will be analytically developed. In the second part of the investigation, a general statement will be established for when OP is constant and r varies.  Finally, in the third part of the investigation, technology will be used to test the two general statements developed in the first two parts, thus determining the general statement for OP’. For this investigation the following technology will be used: TI-Nspire Student Software, Geometer’s Sketchpad, and Microsoft Excel.

Figure 1 shows circle C1with the center O and radius r, and any point P.

image06.png

Figure 1

The circle C2 has center P and radius OP. Let A be one of the points of intersection of C1 and C2. Circle C3has a center A and a radius r. The point P’ is the intersection of C3 with (OP). This is shown in Figure 2, below.

image07.png

Figure 2

It is observed that (OP) is the segment of line between the center of circle C1 and the center of circle C2.

...read more.

Middle

, when r = 1.

Part II.

For Part Two, let OP =2, and an analytic approach will be used to find OP’, when r = 2, r = 3, and r = 4. In this case, OP is the constant, r is the independent variable and OP’ is the dependent variable. Again, the Cosine and Sine Laws will be used to find OP’.

Using TI-Npsire Student Software, the following diagram for when OP=2 and r=2 can be created.

image16.png

Figure 5

        In Figure 5, it is observed that an equilateral triangle has formed, OP= 2 and both OA and AP are also equal to two, since they are both equivalent to r. It can also be observed that point P’ is in the same location as point P, therefore OP = OP’. Thus, OP’ = 2.

When r = 3 and OP = 2, the Cosine Law will be used to find θ.

When r = 3, a =OA=3, b=OP=2, c=AP=2 and C=θ.

cosθ= image01.png

cosθ= image01.png

cosθ= image00.png

cosθ= image00.png

     θ= cos-1 (image00.png

)

Now OAP in ∆AOP is known, and so Φ in ∆AOP’ can be found; since

∠OAP ≅AP’O ≅∠AOP ≅ θ

Since the sum of all angles in a triangle is equal to 180o and ∆AOP’ has two congruent angles, the following can be done:

Φ = 180-2(cos-1(image00.png

))

        Now, we are able to use the Sine Law to find the length of OP’.

image00.png

=image00.png

, where a=OP’=x, b=AO= 2, A=OAP=180-2(cos-1(image00.png

)) and

       B=OP’A= cos-1 (image00.png

)

image02.png

=image02.png

x sin[180-2(cos-1(image00.png

))]= 2sin 180-2(cos-1(image00.png

))

x = image03.png

x = image00.png

OP’ = image00.png

Upon completing all of the calculations, it is established that when

...read more.

Conclusion

 will not exist. OP cannot equal zero (OP≠0) for another reason; looking at the general statement (OP’= image04.png

), OP is the denominator and the denominator can never equal zero because you cannot divide by zero. This has been satisfied by the previous limitation.  As we saw in Part Four, OP must be greater or equal to half the radius {OP

 R I OP ≥image00.png

}, because otherwise circles C1and C2 will not intersect.

Therefore,

Limitations of r: {r

RI 0 < r < OP2}

Limitations of OP: {OP

 R I OP ≥image00.png

}.

Conclusion

The purpose of this investigation was to examine the length of OP’ as it changes with different values of r and OP. Through calculations completed in Part I and Part II, two general statements were formulated. From Part I, where r =1 and OP varied, the statement OP’=image00.png

was derived. From Part II, where OP=2 and r varied, the statement OP’ = image04.png

was analytically determined. From using technology and observing trends in Part III these two general statements were combined into one: OP’= image04.png

. The use of programs such as TI-Nspire Student Software, Excel, and Geometer’s Sketchpad allowed for the efficient collection of great amounts of data. TI-Nspire Student Software and Geometer’s Sketchpad, were also used to create visual representation of the different situations that were presented.The use of these two programs was especially important when checking the validity of the general statement. Upon examining limitations of the general statement it has been concluded that OP’= image04.png

is limited to {OP

 RIOP ≥image00.png

,} and {r

 R I r ≤ OP2}, where OP, r, and OP’ must be Real numbers greater than zero.

...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. Math Studies I.A

    6468.9849 Indonesia 3,900 70.7 275730 15210000 4998.49 Iraq 4,000 69.31 277240 16000000 4803.8761 Isle of Man 35,000 78.49 2747150 1225000000 6160.6801 Italy 31,000 79.94 2478140 961000000 6390.4036 Japan 34,200 82.07 2806794 1169640000 6735.4849 Jordan 5,000 78.55 392750 25000000 6170.1025 Kenya 1,600 55.31 88496 2560000 3059.1961 Korea, North 1,700 71.92 122264

  2. Math IA - Logan's Logo

    To find ?, it was easier to find half of the period first, and then double it to ensure accuracy. Going back to the definition of a period, how long it takes for the curve to repeat itself, it makes sense then that by finding the difference between the x-values

  1. Maths IA Type 2 Modelling a Functional Building. The independent variable in ...

    Therefore, the volume should also increase in proportion to the height of the roof as . Ratio of Wasted Space to Office Block: After calculating the maximum volume of the cuboid, it is important to determine how efficient the volume of the cuboid is by comparing it to the volume of wasted space.

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

    = = r of = 2. We can get the coordinates for . The coordinates for point P will be (2, 0), for it lies on the x-axis and have a radius of 2. As for A, it is still undefined, so we will leave it as the unknown variables, (a, b).

  1. MATH IA- Filling up the petrol tank ARWA and BAO

    ×f2 =600km (5l is for reserve) Arwa fills up her tank at her station along her normal route for US$ p1 per liter. On the other hand Bao drives an extra d kilometers out of his normal route to fill up his vehicle tank for US$ p2 per liter, where p1 is less than p2.

  2. MATH Lacsap's Fractions IA

    To find the sixth row, the row number will be inputted for n and the element will also be inputted for r. The working is shown in Table 6. Table 6: Calculating the values on the sixth row E6 (1)

  1. Music and Maths Investigation. Sine waves and harmony on the piano.

    While a perfectly harmonious chord may sound good to the ear, it doesn?t stand out the way as a slightly off chord may sound. This is unusual, dissimilar and it creates a feeling of uniqueness. To conclude the first part of the exploration, harmonic notes in chords will generally have a simple period ratio to the fundamental note.

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

    I used on the first question using my TI-84 (STAT: CALC: 1-Var Stats: ENTER: List: 2ND: 2: ENTER: Calculate: ENTER) which gives me the following data To find the interquartile range, I would have to use Q3-Q1 = 850-685, it should give me 165.

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