Ib Math Sl Ia - Circles - 2425 Words

Alma Guadalupe Luna Math IA (SL TYPE1) Circles Circles Introduction The objective of this task is to explore the relationship between the positions of points within circles that intersect. The first figure illustrates circle C1 with radius r, centre O, and any point P. r is the distance between the centre O and any point (such as A) of circle C1. Figure 1 The second diagram shows circle C2 with radius OP and centre P, as well as circle C3 with radius r and centre A. An intersection between C1 and C2 is marked by point A. The intersection of C3 with OP is marked by point P’. Figure 2 Through this investigation I will†¦show more content†¦So far, we only know that point P’ lies on the x-axis. This means we only have to assign it one variable, P’(z, 0). We will achieve this by setting the distance formula equal to 1, the total length of AP, and by plugging in A(14, 154) and P’(z, 0). We should end up with 1= (z-14)2+(154)2. We can solve for z to find that z= 12, 0. The zero is discarded because we know by looking at the graph that point P’ isn’t on the origin. P’(z, 0) can now be transformed into P’(12, 0). Because we now know the coordinates of both point P’and point O, we can solve for the length of OP’ through the distance formula. The outcome is that when r is set to a value of 1 and OP is equal to 2, then OP’ will have a total length of 12. If we follow this same process we can form the following table of the length of OP’ when the OP values are changed and r stays at 1. r | OP | OP’ | 1 | 2 | 12 | 1 | 3 | 13 | 1 | 4 | 14 | Through observation we can see that there is an inverse relationship present between the length of OP’ and OP when r is held at a constant value of 1. Thus, we can say that our first general statement will be OP’= 1x, (x= OP). To test the validity of the general statement we can use Geogebra, an application that uses technology with geometric and algebraic software, to check other values. We will first verify that the general statement is valid when OP values are greater than one, for example 16. When r= 1 and OP= 16, OP’=