Part of Final Project
Using these diagrams I will demonstrate the rules of circles that include arcs, chords, secants, and tangents. I recommend hiding whichever circles are not currently being used as this might lead to confusion. Also incase someone would be unaware a chord is line that passes through a circle but we are only interested in that line where it lies in the circle. A secant is a chord but we are usually interested in its properties outside the circle. Finally a tangent is a special case of a secant because it only passes through one point on the circle whereas a secant passes through two. I will begin with circle A. Circle A contains two chords CD and EF. The two sets of vertical angles can be determined by the arc lengths enclosed by their non-central points, for example Angle CPF is equal to ½ (arc CF + arc DE) this also true for its vertical angle pair EPD. This will be true of any two lines that pass through and intersect inside a circle. The lengths of these chords are also related to each other. If you multiply the two parts of a chord it will equal the product of the other chords to parts; for example (CP)(PD)=(EP)(PF). Also if you move either CD and/or EF so that they intersect on the circle rather than in the circle however finding the arc length changes slightly, for this example I will move D so that it is the same as E. Angle CDF (or CEF they are the same in this case) is equal to half the length of arc CF; this angle is called an inscribed angle. Before I move on to the next circle I would like to add that if two chords intersect at A then they are no longer chords, they are diameters and the angles at the intersection of those lines are central angles equal to measure of the arc contained in them. The next circle I will discuss is circle G. This circle also has two chords but they intersect outside the circle. The lengths of these two lines, like the chords, are also related; such that the entire secant multiplied by its exterior part will the equal the same as the other secant. For example (JM)(IM)=(LM)(KM). Also the angle created by the intersection of two secants is relatable to the arcs it contains. In Circle G it would look like this Angle JML = ½(arc JL – arc KI). It looks similar to the chords but as you will notice the two arcs subtract here whereas before they added together. Next I will be moving on to circle T, which has a secant, a tangent, and chord. First I will observe the relationship between the chord and the tangent, since it is the simplest. Like an inscribed angle the angle VA1B1 = ½ arc A1B1. I recommend hiding the chord as we will no longer be using it. The secant and tangent like before have two relationships one involving an angle the other distances. The first is distance where the geometric mean of the entire secant and its outer portion is equal to the tangent: (VA1)2 = (VZ)(VW). The next is angle and arc where the measure of the angle created by the intersection of a tangent and a secant is half the difference of the larger arc and the lesser arc created by those two lines; angle WVA1 = ½(WA1 - ZA1). The final circle N also has two relationships. Circe N has two tangents on it. No matter where you move Q, QR=QS. That is the first and easier of the two relationships involving two tangent lines. The second is that the angle created by the intersection of two tangent lines is equal to half the difference between the larger arc and the lesser arc created by the tangents. In this example it is shown by: angle RQS = 1/2(arc RD1S - arc RE1S). This concludes my explanations of my diagrams in geogebra I hope it was helpful in your understanding of circles and their relation ships to lines that pass through or on them.