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

by rucia (student)

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.

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.

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

This rule is used when we want to know the angles of a triangle. However, there is a limitation for using the rule, which is that the rule is able to be used only all three sides of a triangle are known.

a2= b2+c2-2bcCosA

Problem

Let r=1, find OP’ when OP=2, OP=3, and OP=4.

• When OP=2,

Draw a perpendicular line,

When

=

Link AP’, ∵p’ is on C3,

∴

, ∴

is an isosceles triangle.

∵

is an isosceles triangle,

∴

∴

• When OP=3,

Draw a perpendicular line,

∴ When

=

∵ ...