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

by marsbar1995 (student)

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 C1 with the center O and radius r, and any point P.

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 C3 has a center A and a radius r. The point P’ is the intersection of C3 with (OP). This is shown in Figure 2, below.

Figure 2

It is observed that (OP) is the segment of line between the center of circle C1 and the center of circle C2. As stated above, O is the center of C1 and OP is the radius of C2; therefore C2 always intersects the center of C1. As well, C1 will always intersect the center of C3, since OA is the radius of circle C1 and point A is the center of circle C3.

From the diagram, an isosceles triangle ∆AOP is identified. Since P is the center of C2 and Point A is a point of intersection of circles C1 and C2, AP is the radius of C2. Thus, OP and AP are equal in length because they are both radii of C2. The definition of an isosceles triangle is a triangle with two equal sides; consequently ∆AOP is an isosceles triangle. If a line was drawn from point A to point P’, another isosceles triangles would be formed, ∆AOP’, See Figure 3. Point P’ is the point at which circle C3 intersects with the line which includes points O and P, thus AP’ is a radius of C3. Therefore, AO and AP’ are the same length because they are both radii of C3. Following the definition for an isosceles triangle stated above, ∆AOP’ is an isosceles triangle.

Figure 3

To develop the definition of an isosceles triangle further: The Base Angle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent too. Therefore, OAP is congruent to AOP; and AP’O is congruent to AOP. Refer Figure 4 for a visual representation.

OAP ≈ AP’O ≈ AOP θ

Figure 4

Prior to beginning the investigation it is pertinent to state the number of significant figures that will be used through out. For angle measures, exact values will be kept throughout the calculations. Utilizing exact values for angles will be beneficial because angle measures are used throughout the calculations but do not appear in the final answers; in this way the final answer will be extremely accurate because unnecessary rounding will be avoided. For length measures, using three significant figures will be the most practical and convenient, because length is the quantity that appears in the final answer and three significant figures will allow for easy reading while still keeping a respectable degree of accuracy. When Excel is used, five significant figures will be accepted because this is the standard number of significant figures that Excel presents. Using a constant number of significant digits allows for the same degree of accuracy through out the whole investigation and keeps the investigation orderly. Another important thing to note is that throughout this investigation, all lengths will be measured in Units, since no other unit measure was provided in the parameters for the investigation.

Part I.

For Part One, let r =1, and an analytic approach will be used to find OP’, when OP=2, OP = 3, and OP = 4. In this case, r is the constant, OP is the independent variable and OP’ is the dependent variable. The Cosine and Sine Laws will be used to find OP’.

Cosine Law: c2 = a2 + b2 – 2abcosC         Sine Law:

=

The Cosine Law will be used to find θ.

When OP = 2, a =OA=1, b=OP=2, c=AP=2 and C=θ.

cosθ=

cosθ=

cosθ= ...