Introduction and purpose of task: The purpose of this task is to investigate the positions of points in intersecting circles and to discover the various relationships between said circles. Circle C1 has center O and radius r. Circle C2 has center P and radius OP. Let A be one of the points of intersection of C1 and C2. Circle C3 has center A and radius r (therefore circles C1 and C3 are the same size). The point P’ (written P prime) is the intersection of C3 with OP. This is shown in the diagram below.

Analytically find OP’ using r=1 and OP=2, OP=3, and OP=4:
First, I created a line (see the dashed line in the above figure) between AP’ that creates the ΔAOP’. Because P’ is on the circumference of circle C3 and A is the center of circle C3, that means that AP’ is equal to the radius of C3, which is 1. We also know that because line AO connects the circumference of C1 with the center of C1 (O) and the circumference of C3 with the center of C3 (A), the radii of these circles is the same, which means that they are equivalent circles. Therefore, in the ΔAOP’, AO=AP. When a triangle has two equivalent sides, it is an isosceles triangle. By that logic, ∠O=∠P’. Now, I looked at the triangle that is already drawn in the above figure, ΔAOP. We know that this triangle is also isosceles because OP=AP. By that logic, ∠A=∠O.

Using the law of cosines c^2=a^2+b^2-2abcos(C), which works for any triangle, I assigned θ to ∠O and determined that cos(θ)=1/(2*OP). Then, using the law of sines (insert law of sines here), sin(θ)/1=sin(180-2θ)/OP’ OP’=sin(180-2θ)/sin(θ)

OP’=sin(2θ)/sin(θ)
OP’=2cos(θ)
But because cos(θ)=1/2OP as earlier discovered;
OP’=1/OP

By using this equation, I derived the following answers analytically using r=1 and OP=2, OP=3, and OP=4. OP234
OP'0.50.330.25

Behavior of intersecting circles and general statement describing interaction that occurs when value of OP is changed: As OP changes, the resulting OP’ value...

...Santoro, A. (2004). “Manipulatives: A Hands-On Approach to Math.” Principal, 84 (2), (28-28).
This article speaks about the importance and significance of the use of manipulatives in the classroom, specifically in the subject of math. Manipulatives have proven to be valuable when used in a math class and are even more valuable to the children when they are young, and are learning new math concepts. Students are able to physically...

...MathIAMath Internal Assessment
EF International Academy NY Student Name: Joo Hwan Kim Teacher: Ms. Gueye Date: March 16th 2012
Contents
Introduction Part A Part B Conclusion
Introduction
The aim of this IA is to find out the pattern of the equations with complex numbers by using our knowledge. I used de Moivre’s theorem and binomial expansion, to find out the specific pattern and make conjecture about it. I basically used...

...Name: Linh Nguyen
IB MathIA
02/06/12
Part A
Consider this 2× 2 system of linear equations: x + 2y = 3
2x - y = -4
We can see patterns in the constants of both equations. In the first equation, the constants are 1, 2, and 3. The common difference between the constants is 1:
3 – 2 = 1
2 – 1 = 1
Based on this, we can set up a general formula for the constant of this equation:
Un = U1 + (n - 1)d Where:
n: The number of the series
d: the...

...IB Math Studies Internal Assessment:
Is the distance a tennis ball travels horizontally dependent on the angle of which it is dropped at?
Exam Session: May 2014
School Name:
Teacher:
Course: IB Math Studies
Word Count: 654
Name:
Is the distance a tennis ball travels horizontally dependent on the angle of which it is dropped at?
Introduction
In tennis, players hit the tennis ball in certain ways so the ball goes the way they want it to...

...MathIA
IB MATH SL
MathIA
Introduction: In this task I will consider a set of numbers that are presented in a symmetrical pattern and try to find a general equation to find the elements in the [pic]row.
Consider the five rows of number shown below.
Figure 1 Lacsap’s Fractions
The aim of this task is to find the numerator of...

...object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
5....

...• What were the most revolutionary social and economic developments of the last quarter of the nineteenth century?
• How did different groups of Americans respond to those changes and how effective were their responses?
• What role did government play in these developments?
In the late 1900s some of the most social and economic developments were railroads, steel oil, the type writer cash register, light bulb and agriculture. The development of the railroad made it easier for immigrant to...

...
ANALYSIS
Physics has a lot of topics to cover. In the previous experiments, we discussed Forces, Kinematics, and Motions. In this experiment, the focus is all about Friction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction like fluid friction which describes the friction between layers of a viscous fluid that are moving relative to each other; dry friction which...