# Ib Sl 2012-2013 Math Circles Portfolio

Topics: Triangle, Law of cosines, Pythagorean theorem Pages: 3 (1074 words) Published: February 25, 2013
OPA Circle Style

The three circles; O, P and A intersect to create an interesting investigation regarding circles. Since this is a Calculus course, the investigation does have to deal with Derivatives. The most important and the focus of this portfolio is the line segment, OP’. Using the given diagram above, this investigation consists of finding the general equation for discovering OP’.

The values that are given are that r, is the radius of C1 and C2. OP and AP are the radii of C3. This information allows for the information to be manipulated too create two isosceles triangles. The first triangle and the one that is given, ∆OPA is an isosceles triangle therefore it can be concluded, thanks to the Isosceles Triangle Theorem that angle O and A are congruent to each other in this triangle. ∆OPA is not the only triangle that can be created, ∆OP’A is the second triangle created with a radius from C2. Therefore ∆OP’A is also an isosceles triangle. Now in both the triangles stated above, they share a common angle, O. With the help of this information we can analyze the preliminary relationship between OP and OP’. We can find OP’, we have the tools. The two tools that we will be using mainly are the sine law and cosine law (respectively);

Sin Aa = Sin Bb = Sin Cca2= b2+ c2-2bc cosA

and the two triangles that are going to be used;
For the first calculations, r=1 and OP =2. By finding the ∠O in one triangle, I have found the ∠O in both triangles, allowing me a complete ration to perform the sine law. Side Note: All Final Answers are rounded to 3 Significant Figures. For the first calculations, r=1 and OP =2. By finding the ∠O in one triangle, I have found the ∠O in both triangles, allowing me a complete ration to perform the sine law. Side Note: All Final Answers are rounded to 3 Significant Figures. When a triangle has all three sides given and an angle needs to found, the Cosine Law can be used. By finding angle O in ∆OPA, a complete ratio...