# Math SL Circle Portfolio. The aim of this task is to investigate positions of points in intersecting circles.

Math IA (SL TYPE I)

Circles

Jennifer

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

Introduction

The following diagram shows a circle  with centre O and radius r, and any point P.

In this case, the r is the distance between any point (such as A) and the centre point O of the circle . Since the radius is 1 unit, OP will also be 1 unit away from the point O.

The circle  has centre P and radius . A is one of the points of intersection of  and . Circle  has centre A, and radius r. The point P’ is the intersection of  with (OP). The r=1, =2, and P’=0.5. This is shown in the diagram below.

This investigation will explore in depth of the relationship between the r value and  values, when r is held constant and  values modified. It will also investigate the reverse, the relationship when the   values are held constant and the r values are modified.

In the first example, the r value given is 1. An analytic approach will be taken to find the length of   when =2. Firstly, one can note that 2 isosceles triangle can be drawn by using the points A, O, P’, and P. It is shown in the diagram below.

In ΔAOP’, lines  and  have the same length, because both points, O and P’ are within the circumference of the circle , which means that  and   are its radius. Similarly, ΔAOP forms another isosceles triangle, because the lines   and   are both radii of the circle .

= r of or = 1. Since the circles are all graphed, they can be given coordinates. For , the coordinates of the point O will be (0, 0), because it lies in the origin of the graph. The coordinates of the point A is undefined, so we will substitute in variables for its values, (a, b). We have to find the coordinate of point A to find the coordinates of the point P’.

= = 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).

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)

Distance formula: 2=

(2=

4=

Through the system of equations, the value of ‘a’ from the coordinates of point A can be found:

1= +

•   4=

________________________________

1= 4a

a=

Then, through substitution, the value of ‘b’ can be found:

1= +

1=  +

= 1-

= ...