'Ia Math\r'

'IA Task I Introduction and bearing of task: The purpose of this task is to investigate the positions of points in cross circles and to discover the various relationships between state circles. Circle C1 has subject matter O and gas constant r. Circle C2 has center P and radius OP. Let A be wiz of the points of crossway of C1 and C2. Circle C3 has center A and radius r (therefore circles C1 and C3 are the alike(p) size). The point P’ (written P prime) is the intersection of C3 with OP. This is shown in the diagram below.analytically find OP’ exploitation r=1 and OP=2, OP=3, and OP=4: First, I created a guide (see the dashed strain in the above figure) between AP’ that creates the ? AOP’. Because P’ is on the circumference of circle C3 and A is the center of circle C3, that means that AP’ is represent to the radius of C3, which is 1. We also last that because line AO connects the circumference of C1 with the center of C1 (O) and t he circumference of C3 with the center of C3 (A), the radii of these circles is the same, which means that they are eq circles.Therefore, in the ? AOP’, 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, ? AOP. 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 assign ? to ? O and determined that cos(? )=1/(2*OP). Then, apply the law of sines (insert law of sines here), sin(? )/1=sin(180-2? )/OP’ OP’=sin(180-2? /sin(? ) OP’=sin(2? )/sin(? ) OP’=2cos(? ) nevertheless because cos(? )=1/2OP as earlier sight; 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 OP0. 50. 330. 25 Behavior of encounter circles and commonplace account describing interaction that occurs when grade of OP is changed: As OP changes, the resulting OP’ value decreases exponentially. This shows that ______________ harangue BLAH BLAH BLAH BLAH Analytically find OP’ using OP=2 and r=2, r=3, and r-4:Behavior of intersecting circles and general statement describing interaction that occurs when value of OP is changed: Comments on consistency of general statement #1 vs. general statement #2: exercising technology to investigate other determine of r and OP. Find the general statement for OP’: Test the validity of the most youthful general statement by using different values of OP and r: handling of scope and/or limitations of the most new-fangled general statement: Explanation of how general statement was arrived at: further comments: Further investigation into relationships of intersecting circles:\r\n'