Slide 1 / 255 Slide 2 / 255 Geometry Circles 2015-10-23 www.njctl.org Slide 3 / 255 Slide 4 / 255 Table of Contents Throughout this unit, the Standards for Mathematical Practice Click on a topic to go are used. to that section MP1: Making sense of problems & persevere in solving them. Parts of a Circle MP2: Reason abstractly & quantitatively. Central Angles & Arcs MP3: Construct viable arguments and critique the reasoning of others. Arc Length & Radians MP4: Model with mathematics. MP5: Use appropriate tools strategically. Chords, Inscribed Angles & Triangles MP6: Attend to precision. MP7: Look for & make use of structure. Tangents & Secants MP8: Look for & express regularity in repeated reasoning. Segments & Circles Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on Questions from Released PARCC Examination this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 5 / 255 Slide 6 / 255 Circles are a type of Figure A figure lies in a plane and is contained by a boundary. Parts of a Circle Euclid defined figures in this way: Definition 13: A boundary is that which is an extremity of anything. Definition 14: A figure is that which is contained by any boundary or boundaries. A boundary divides a plane into those parts that are within the boundary and those parts that are outside it. That which is within the boundary is the "figure." Return to the table of contents
Slide 7 / 255 Slide 8 / 255 Circles Circles and Their Parts Another way of saying this is that a circle is made up of all Euclid defined a circle and its center in this way: the points that are an equal distance from the center of the Definition 15: A circle is a plane figure contained by one line circle. such that all the straight lines falling upon it from one point among those lying within the figure equal one another. radius Definition 16: And the point is called the center of the circle. center This states that all the radii (plural of radius) drawn from the center of a circle are of equal length, which is a very circumference important aspect of circles and their radii. Slide 9 / 255 Slide 10 / 255 Radii Circles and Their Parts A radius (plural, radii ) is a line segment drawn from the center of the circle to any point on its circumference. The symbol for a circle is and is named by a capital letter placed by the center of the circle. It follows from the definition of a circle that all the radii of a A circle are congruent since they must all have equal length. The below circle is named: Circle A or An unlimited number of radii can be drawn in a circle. A AB is a radius of That all radii of a circle are congruent will be important to A radius (plural, radii ) is a line solving problems. C segment drawn from the center of the A A circle to any point on the circle. In this drawing, we know that line segments AC, AD and AB B B are all congruent. D Slide 11 / 255 Slide 12 / 255 Diameters Diameters Definition 17: A diameter of the circle is any straight line drawn There are an unlimited number of diameters which can be drawn through the center and terminated in both directions by the within a circle. circumference of the circle, and such a straight line also bisects They are all the same length, so they are all congruent. the circle. Since the diameter passes through the center of the circle and That all diameters of a circle F extends to the circumference on either side, it is twice the length E are congruent will be important of a radius of that circle. to solving problems. C A In this drawing, we know that G line segments BE, CG and DF B are all congruent. D
Slide 13 / 255 Slide 14 / 255 Chords Chords A chord is a line segment whose endpoints lie on the There are an unlimited number of chords which can be circumference of the circle. drawn in a circle. So, a diameter is a special case of a chord. Chords are not necessarily the same length, so are not necessarily congruent. Why is a radius not a chord? B B D D F F Chords can be of any length up to a All the line segments in this drawing are chords. maximum. A A What is the longest chord that can be G G drawn in a circle? E E C C Slide 15 / 255 Slide 16 / 255 Semicircles Diameters and R adii The measure of the diameter, d , is twice the measure of Definition 18: A semicircle is the figure contained by the the radius, r . diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. In this case, CD = 2 AB In general, d = 2r or r = 1/2 d diameter Example: In the diagram to the semicircles D left, AB = 5. Determine AC & DC. A B C Slide 17 / 255 Slide 18 / 255 1 A diameter of a circle is the longest chord of the circle. 2 A radius of a circle is a chord of a circle. True True False False
Slide 19 / 255 Slide 20 / 255 3 The length of the diameter of a circle is equal to twice the 4 If the radius of a circle measures 3.8 meters, what is length of its radius. the measure of the diameter? True False Slide 21 / 255 Slide 22 / 255 5 How many diameters can be drawn in a circle? A 1 B 2 Central Angles C 4 & Arcs D infinitely many Return to the table of contents Slide 23 / 255 Slide 24 / 255 Arcs Arcs A Arc AB or AB A Arc AB or AB B Of course, you may have wondered B why we went around the circle the way we did (in blue). Arc AB is the path between points A This shorter path (blue) is called the and B on the circumference of the minor arc. We use 2 letters to circle. C denote minor arcs The longer path (green) is called the major arc. If we want to refer to a major arc, we will add another point along that path and include it in the name. For instance, we would name the green arc, Arc ACB to distinguish it from Arc AB.
Slide 25 / 255 Slide 26 / 255 Arcs Central Angles A A Arc AB or AB A central angle of a circle is any angle which has vertices consisting B B of the center of the circle and two points on the circumference. O Now let's discuss some ways we can use arcs, and measure them. How many different central angles can you find in this diagram? D C C Name them. Slide 27 / 255 Slide 28 / 255 Adjacent Arcs Central Angles A A Adjacent arcs : two arcs of The measure of a central angle is the same circle are equal to the measure of the arc it adjacent if they have a B intercepts. common endpoint. 70 º O So, the measure of the angle AOB is In this case, Arc CA and the same as the measure of Arc AB, Arc AT are adjacent since C T O they share the endpoint A. also denoted by mAB. C In this case, the mAB is 70º, since that is the m ∠ AOB. That also means the mACB is 290º, since a full trip around the circle is 360 º. Slide 29 / 255 Slide 30 / 255 Adjacent Arcs 6 Find the measure of Arc RU. A From the Angle Addition Postulate T we know that S m ∠COT = m ∠COA + m ∠AOT 30º It follows then that the measures 90º C T of adjacent arcs can be added to 100º O U find the measure of the arc formed 80º by the adjacent arcs. 60º In this case that: R mCAT = mCA + mAT V which is the Arc Addition Postulate
Slide 31 / 255 Slide 32 / 255 7 Find the measure of Arc RT. 8 Find the measure of Arc RVT. T T S S 30º 30º 90º 90º 100º U 100º U 80º 80º 60º 60º R R V V Slide 33 / 255 Slide 34 / 255 9 Find the measure of Arc UST. 10 Which type of arc is Arc TQR? A Minor Arc B Major Arc T C Semicircle S D None of these 30º 90º T 100º U Q 80º 120º 60º 60º 80º R V Note that you need to use the R indicated degree measures as the S drawing is not to scale. Slide 35 / 255 Slide 36 / 255 11 Which type of arc is Arc QRT? 12 Which type of arc is Arc QS? A Minor Arc B Major Arc A Minor Arc C Semicircle T Q B Major Arc D None of these 120º C Semicircle 60º 80º T Q D None of these 120º R S 60º 80º Note that you need to use the Note that you need to use the R indicated degree measures as the indicated degree measures as the S drawing is not to scale. drawing is not to scale.
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