Circle Theorems
Circle theorems are a set of rules and principles that apply to circles and their properties. They are used to solve problems involving angles, chords, tangents, and other geometric elements within circles. These theorems are fundamental in geometry and are essential for understanding and solving problems related to circles.
5 Key excerpts on "Circle Theorems"
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...All of the other properties listed are true. Circles The last of the plane (two-dimensional) figures we will look at is the circle, an important shape indeed. Just look around you, and you will see many circular objects. It is important in life, as well as on the GED ® test, that you know some facts about circles. A circle is defined as all the points at a fixed distance from a certain point. That certain point is called the center of the circle, which is usually designated by an O. The fixed distance from the center to the circle is called the radius of the circle. If we extend the radius across the center to the other side of the circle, we get the diameter, which is just twice the radius. The circle is usually named by its center, so this is circle O. Another important value when talking about circles is the Greek letter pi(π), which is the ratio of the circumference (perimeter) of the circle to its diameter. Pi has a value of or roughly 3.14. The GED ® test will indicate what form to use for an answer, but it usually is in terms of π. C ALCULATOR The calculator has a button labeled π, and you can treat π just as you would any constant. That is, to total 2 π + 3 π, you can press those buttons, with “enter” and the value 5 π appears. If you then use the toggle button, the calculator will give the decimal equivalent, but that usually isn’t required. The angle with the center of a circle at its vertex and two radii (plural of radius) as its sides is called a central angle. The sum of the central angles in a complete rotation (or circle) is 360°. Think of the hands of a clock. At 3:00, the angle between the hands on a clock is 90° (one-fourth of the way around the whole clock face). It is also 90° at 9:00. The portion of the circumference of a circle between two points is called an arc. Its measurement is the same as the measure of the central angle drawn to it (see the figure below). Be careful when figuring the angle between the hands of a clock...
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Chapter 9 Circles Your Goals for Chapter 9 1. You should be able to identify the parts of a circle. 2. You should be able to determine the measure of arcs and the angles created in a circle. 3. You should be able to solve problems using circumference, arc length, and areas of circles and sectors. 4. You should be able identify the center and radius of a circle and graph the circle on a coordinate plane from an equation in center-radius form. 5. You should be able to write an equation of a circle given the center and radius. Standards The following standards are assessed on Florida’s Geometry End-of-Course exam either directly or indirectly: MA.912.G.6.2: (Low) Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles. MA.912.G.6.4: (Moderate) Determine and use measures of arcs and related angles (central, inscribed, and intersections of chords, secants and tangents). MA.912.G.6.5: (High) Solve real-world problems using measures of circumference, arc length, and areas of circles and sectors. MA.912.G.6.6: (Moderate) Given the center and the radius, find the equation of a circle in the coordinate plane or given the equation of a circle in center-radius form, state the center and the radius of the circle. MA.912.G.6.7: (Moderate) Given the equation of a circle in center-radius form or given the center and the radius of a circle, sketch the graph of the circle. Lines and Segments A radius is a segment whose endpoints are the center and any point on the circle. A chord is a segment whose endpoints are on the circle. A diameter is a segment whose endpoints are on the circle and passes through the center. A diameter is the longest chord of the circle. A secant is a line that intersects the circle at two points. A tangent is a line that intersects the circle at exactly one point. Example: Identify each of the following in the given circle. A. chord B. secant C...
- Alfred Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...You know that all the angles are 90 degrees because of Thales’ Theorem. The proof of this theorem is relatively simple and can be done as follows: Select any point, P, on the semicircle, and draw and, as shown in figure 4.8. Because,, are all radii of the same circle, we have two isosceles triangles, POA and POC. Thus their base angles are congruent. Using algebra, in triangle PAC, ∠1 + (∠1 + ∠2) + ∠2 = 180° 2 · (∠1) + 2 · (∠2) = 180° ∠1 + ∠2 = 90° and therefore the theorem is proved. Of course, once the class has learned that an inscribed angle contains one-half the number of degrees as its intercepted arc, they know that all the angles are right angles since arc ADC is 180 degrees. Topic: Introducing the Nature (or Importance) of Proof Materials or Equipment Needed A sheet with the list shown below. 3 = 2 0 + 2 5 = 2 1 + 3 7 = 2 2 + 3 9 = 2 2 + 5 Implementation of the Motivation Strategy When a teacher embarks on the concept of doing a proof in mathematics, students often see this activity as another mathematical process they are required to learn without understanding the significance of proving that something is true for all cases. To motivate students, the teacher should demonstrate that you cannot just assume something is true because it appears that way. Begin the lesson by asking students if they believe that the following statement is true. Every odd number greater than 1 can be expressed as the sum of a power of 2 and a prime number. They should justify their response. Typically, students will try to see if this statement holds true for the first several cases. In the short time allowed at the start of the lesson, this will probably suffice for the students to conclude that this is a true statement...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...In fact, these items are right circular cylinders; they may be very short cylinders, but they are not figures in a plane and they have more dimensions than do circles. A circle is a point and every member of the set of points is the exact same distance from the center in the plane. The circular array of points is the circumference of the circle. The circle is the ring. The set of points inside the ring is not the circle; it is the interior of the circle. Similarly, the points outside the ring are not the circle; they make up the exterior of the circle. A circle has no substance, so the best way to model a circle is with a wire ring. Of course, the wire of the ring has thickness, but it represents a much better model of a circle than a solid plastic disk. There are several important terms associated with circles, some of which are shown in Fig. 4.18. The diameter is a segment that joins two points on the circle and passes through the center. A radius is half a diameter and goes from the center to any point on the circumference. A chord is a line segment that joins any two points on a circle. The diameter is the longest chord of a circle. A few additional terms are shown in Fig. 4.19. A sector of a circle is a pie-wedge region bounded by two radii and an arc. A segment is a region bounded by a chord and an arc. Fig. 4.19. Your Turn 16. Use the information in Fig. 4.18 and Fig. 4.19 to write your own informal definitions for the following terms: a) Center b) Chord c) Circle d) Circumference e) Diameter f) Radius g) Sector h) Segment Constructions Classic constructions, completed with nothing more than a compass and straightedge (not a ruler), are elegant and beautiful in form. However, the basic concepts of construction can be introduced using paper folding. For this, you need a straightedge, a pencil, and some paper. Waxed paper, tracing paper, or meat patty paper are best, because these are thin and show the creases of the construction well...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...The distance of point P to the diagrammed circle with center O is the line segment, part of line segment. A line that has one and only one point of intersection with a circle is called a tangent to that circle, and their common point is called a point of tangency. In the diagram, Q and P are each points of tangency. A tangent is always perpendicular to the radius drawn to the point of tangency. Congruent circles are circles whose radii are congruent. If O 1 A 1 ≅ O 2 A 2, then O 1 ≅ O 2. Circles that have the same center and unequal radii are called concentric circles. A circumscribed circle is a circle passing through all the vertices of a polygon. The polygon is said to be inscribed in the circle. PROBLEM A and B are points on a circle Q such that is equilateral. If the length of side = 12, find the length of arc AB. SOLUTION To find the length of arc AB, we must find the measure of the central angle AQB and the measure of radius. AQB is an interior angle of the equilateral triangle. Therefore, m AQB = 60°. Similarly, in the equilateral Given the radius, r, and the central angle, n, the arc length is given by Therefore, the length of arc AB = 4π. FORMULAS FOR AREA AND PERIMETER Figures Areas Area (A) of a: square A = s 2 ; where s = side rectangle A = lw ; where l = length, w = width parallelogram A = bh ; where b = base, h = height triangle A bh ; where b = base, h = height circle A = πr 2 ; where π = 3.14, r = radius sector A = ; where n =. central angle, r = radius, π = 3.14 trapezoid A = (h)(b 1 + b 2); where h = height, b 1 and b 2 = bases Figures Perimeters Perimeter (P) of a: square P = 4 s ; where s = side rectangle P = 2 l + 2 w ; where l = length, w = width triangle P = a + b + c ; where a, b, and c are the sides Circumference (C) of a circle C = πd ; where π = 3.14, d =. diameter PROBLEM Points P and R lie on circle Q, m PQR = 120°, and PQ = 18...
