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Mathematic SL IA -Circles (scored 17 out of 20)

Extracts from this document...

Introduction

Mathematics Standard Level Portfolio

Type 1- Circles

Candidate name: Sun Ha (Rucia). Park

Candidate number:

School: Beijing No.55 High School


INTRODUCTION

        In this task, I will investigate positions of points in intersecting circles.

I will analyze and investigate the problem in two different situations with different conditions, to find a general statement that shows general situation of points in intersecting circles. I am going to use Microsoft word to analyze the task, an application called Geogebra to graph intersecting circles, and TI-84 graphing calculator to calculate the values.

There are three circles intersecting. Circle 1(C1) has center O, Circle 2 (C2) has center P and Circle 3(C3) has center A. C1 has radius OP, and let A be the intersecting point of C1 and C2. C3 has its radius, r. The point P’ is the intersection of C3 with OP.

image21.jpg

Used knowledge:

        I have used 3 mathematical theorem, or rules to proof my findings.

  1. Similar triangle theorem

This theorem is used where two or more triangles with the same size angles. If the triangles are in the same shape with the same three (in fact, two) angles, then they are called similar triangles in different ratio.

image22.jpg

  1. The Pythagoras’s theorem

Sides which are opposite to the angles are labeled using small alphabets of the angles. When angle B is 90°, which means the triangle is a right triangle, then

a2 +c2=b2.

  1. Cosine rule

...read more.

Middle

We can notice that the length of OP’ gets small as the length of OP gets longer. Be specific, we can know that the length of OP’ is the reciprocal of the length of OP which is that OP’ is image06.png

 , when OP is 2. From here, we can set a general statement, OP’=image06.png

 . However, we have to realize that the length of r is 1, and this means the numerator of OP’ is not a just form of reciprocal number.  Therefore, the general statement can be interpreted to another form, OP’=image06.png

 .

Let’s consider another condition.

In the second condition, let OP=2, fixed, and find OP’ depending on the different lengths of r.

image31.jpg

  • when r=2,

By the graph I’ve drawn, we can know that OP’ is on the same point of OP.

Therefore, image02.png

  • When r=3, image32.jpg

As the cosine rule,

image09.png

image00.png

image02.png

image06.png

Link

image10.png

 is an isosceles triangle, and

is an isosceles triangle

image02.png

image06.png

image11.png

image06.png

image12.png

image13.png

  • When r=4,

image33.jpg

As the graph shown,

∵AO = 4 = r,

OP’ = 2AO = 2r

∴OP’ = 8

From the second assumption, we can find;

OP

2

2

2

r

2

3

4

OP’

2

image06.png

8

As the length of OP is stable and the length of r is changeable, the length of OP’ gets longer, which means the length of OP’ changes only by the changes in the length of r. In the first concept, the general statement I gave was OP’=image06.png

 , however, it doesn’t match in this condition. If we follow the first general statement, the length of OP’ should be 1 when r=2. The same for the others, it should be image06.png

, and 2 when r=3, and r=4 respectively. Let’s make a diagram shown clearly.

Length of OP’ following the first general statement in the second condition.

1

image06.png

2

Actual length of OP’

2

image06.png

8

From the diagram, we can figure out that the actual length of OP’ is calculated by multiplying one more time of r to the length of OP’ calculated which is followed the first general statement. Therefore, we can approach to the new general statement, which is OP’=image14.png

 .

To test the validity of my general statement, I would like to change the values of OP and r. I will check up for the validity by using FOUR different conditions;

        Before testing, I am going to estimate the results using the general statement. The lengths of OP’ are going to be 1, image06.png

, image06.png

and 9 respectively.

Length of OP'

Condition

Hypothetic value (by using the general statement, OP’=image14.png

)

Real value

1

?

image06.png

?

image06.png

?

9

?

...read more.

Conclusion

1 and C2 don’t intersect at one point when OP=image06.png

r. This tells us about the length of OP should be longer than image06.png

r. I assume one more situation which has the length of OP is 1 and the length of r is 2. In this situation, OP=image06.png

r. I tried to draw the circles, and it was possible to draw three circles.

Now, what I have is that the length OP should be longer than image06.png

r, but it’s possible to equal image06.png

r. Then, let K denote the coefficient of r, set OP=Kr and test what value of K would be, whether three circles can be drawn. I would like to find the value of K which is greater than image06.png

 and smaller than image06.png

 , image06.png

. I assume a situation which is OP of 5, r of 12.

image26.jpg

The graph shows, if K=image06.png

, so OP=image06.png

r, then C1 and C2 couldn’t intersect at one point, therefore, it is impossible to draw three circles required.

image27.jpg

Since OP and r are the radii of circles, so their values can’t be negative. Moreover, the length of OP should not be less than image06.png

r, the limitation of the general statement is image06.png

r.

        As conclusion, I could find out the general statement by two different conditions which have different fixed value of OP and r, and testing the validity of the general statement I found. Thus, the general statement I found finally is OP’=image14.png

. However, the statement also has a scope, which is image06.png

r.

...read more.

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