Inscribed Angles

5 Key excerpts on "Inscribed Angles"

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  eBook - ePub

  Solving Problems in Geometry

  Insights and Strategies for Mathematical Olympiad and Competitions

  • Kim Hoo Hang, Haibin Wang;;;(Authors)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  Chapter 3

  Circles and Angles

  A circle is uniquely determined by its center and radius, i.e., if two circles have the same center and radius, they must coincide. We use O to denote a circle centered at O.

  It is widely known that given a circle with radius r, its perimeter equals 2πr and the area of the disc is πr2 . Indeed, there are many more interesting properties about circles. In this chapter, we will focus on the properties of angles related to circles.

  3.1 Angles inside a Circle

  Theorem 3.1.1 An angle at the center of a circle is twice of the angle at the circumference.

  Proof. Refer to the diagram below. We are to show ∠BOC = 2∠BAC. Extend AO to D. Since O is the center of the circle, we have AO = BO. Now ∠B = ∠OAB in ΔAOB, and the exterior angle ∠BOD = ∠B +∠OAB = 2∠OAB.

  Similarly, ∠COD = ∠C +∠OAC = 2∠OAC.

  Now ∠BOC = ∠BOD +∠COD = 2∠OAB + 2∠OAC = 2∠BAC.

  Notice that the proof is not completed yet: there is another possible situation as illustrated in the diagram on the right. Notice that the proof above does not apply in this situation, but an amended version following the same idea (using subtraction instead of addition) leads to the conclusion. We leave it to the reader.

  Example 3.1.2 Let O be the circumcenter of ΔABC. We have:

  (1) ∠BOC = 2∠A

  (2) ∠OBC = 90°−∠A

  Proof. (1) follows directly from Theorem 3.1.1 .

  Theorem 3.1.1 has a few immediate corollaries which are very important in circle geometry.

  Corollary 3.1.3 Angles in the same arc are the same.

  Refer to the left diagram below. ∠1=∠2 because they are both equal to half of the angle at the center of the circle.

  We call a quadrilateral cyclic if it is inscribed inside a circle.

  Corollary 3.1.4 Opposite angles of a cyclic quadrilateral are supplementary, i.e., their sum is 180°.

  Refer to the previous right diagram. We have ∠1+∠2

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  Further Mathematics

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  • If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle. • If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. • If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental. o For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle. • An inscribed angle subtended by a diameter is a right angle. • The diameter is the longest chord of the circle. Sagitta ________________________ WORLD TECHNOLOGIES ________________________ • The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. • Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines: Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x , since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2 r − x ) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2 r − x ) x = ( y /2)². Solving for r , we find the required result. Tangent • The line drawn perpendicular to a radius through the end point of the radius is a tangent to the circle. • A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle. • Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  ∠ 1 m ∠ ABE m ∠ 2 m ∠ ABD m ∠ 1 m DE 84 — AC ! DB O m ∠ 1 1 2 m ABC . m ABC 180 ABC m ∠ 1 90 ∠ 1 AC CD ! m ∠ 1 1 2 m ABC CD ! CA 282 CHAPTER 6 ■ CIRCLES Unless otherwise noted, all content on this page is © Cengage Learning. COROLLARY 6.2.4 The measure of an angle formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc. (See Figure 6.29 on page 281.) 1 2 O A D E B C E 1 A D B C Figure 6.30 Figure 6.31 General Rule: With the help of an auxiliary line, Theorems 6.2.5, 6.2.6, and 6.2.7 can be proved by using Theorem 6.1.2 (measure of an inscribed angle). Illustration: In the proof of Theorem 6.2.5, the auxiliary chord places in the position of an exterior angle of . BCD ∠ 1 BD STRATEGY FOR PROOF ■ Proving Angle-Measure Theorems in the Circle THEOREM 6.2.5 The measure of an angle formed when two secants intersect at a point outside the circle is one-half the difference of the measures of the two intercepted arcs. GIVEN: Secants and as shown in Figure 6.31 PROVE: PROOF: Draw to form . Then the measure of the exterior angle of is given by , so Because and are Inscribed Angles, and . Then or . m ∠ C 1 2 (m AD m BE ) m ∠ C 1 2 m AD 1 2 m BE m ∠ D 1 2 m BE m ∠ 1 1 2 m AD ∠ D ∠ 1 m ∠ C m ∠ 1 m ∠ D . m ∠ 1 m ∠ C m ∠ D BCD BCD BD m ∠ C 1 2 (m AD m BE ) DC AC EXS. 11, 12 Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. NOTE: In an application of Theorem 6.2.5, one subtracts the measure of the smaller arc from the measure of the larger arc. EXAMPLE 4 GIVEN: In of Figure 6.32, and FIND: SOLUTION If , then .

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  Geometry

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2004(Publication Date)
  • Wiley

    (Publisher)

  C = πd Divide both sides of the equation by π. = d Now we have an equation for the diameter of a circle. = d We’ll substitute the values for C and π. d ≈ 5.1 inches The symbol we’ll use for a circle is . A chord is a line joining any two points on the circumference. Thus AB and AC are chords of the following circle. C A B O 16 3.14 C π 76 GEOMETRY An arc is part of the circumference of a circle. The symbol for arc is . Thus AB refers to arc AB. An arc of 1° is th of a circle. It’s important to distinguish between chords and arcs named with the same letters. In the previous circle, chord AB is a straight line joining points A and B, which are both on the circumference of the circle. But arc AB is actually the curved part of the circumference running from point A to point B. A diameter is a chord that runs through the center of the circle; it’s the longest possi- ble chord. Thus in the last figure, AC is the diameter of O. The diameter divides a circle into two semicircles. A semicircle is an arc equal to one-half of the circumference of a circle; a semicircle contains 180°. Example 6: Draw a chord, CD, on M. Solution: Here are two of the many possible chords. Are you ready for some congruent circles? Congruent circles are circles having con- gruent radii. Thus if OE = O′G′, then circle O circle O′. O G' F E O' M C D M C D M 1 360 Circles 77 Here’s one last term: central angle. A central angle is an angle formed by two radii. In the following figure, the angle between radii OB and OC is a central angle. Therefore, the central angle between OB and OC is O, which can also be writ- ten BOC. Sometimes angles are not represented in degrees, but in units called radians. If a central angle of a circle intercepts an arc equal in length to the radius of the cir- cle, the central angle is defined as 1 radian. Because the radius can be marked off along the circumference 2π (or about 6.28) times, we see that 2π = 360°, or π = 180°.

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  Reeds Vol 1: Mathematics for Marine Engineers

  • Kevin Corner, Leslie Jackson, William Embleton(Authors)
  • 2013(Publication Date)
  • Thomas Reed

    (Publisher)

  7 TRIGONOMETRY AND GEOMETRY An angle is the corner of two joining lines and the magnitude of an angle is measured in either degrees or radians. Measurement of Angles The most common unit of angle measurement that is known is probably the degree . There are 360 ◦ in a circle and so a degree is one three-hundred-and-sixtieth part of a circle. The symbol for a degree is ◦ and so there are 360 ◦ in a circle. Figure 7.1 shows a quarter of a circle which is 90 ◦ and is termed a right-angle , an angle which is less than 90 ◦ (Figure 7.2) is called an acute angle, greater than 90 ◦ but less than 180 ◦ (Figure 7.3) is an obtuse angle, and greater 180 ◦ (Figure 7.4) is a reflex angle. Right angle 90° Figure 7.1 142 • Mathematics Acute angle <90° Figure 7.2 90° < Obtuse angle < 180° Figure 7.3 Reflex angle >180° Figure 7.4 One sixtieth part of a degree is termed one minute and the sixtieth part of a minute is one second . Symbols are used to represent degrees, minutes and seconds. 60 s = 60 = 1 min; 60 min = 60 = 1 ◦ ; 360 ◦ = 1 circle. An angle of 35 degrees 23 minutes and 15 seconds is written as 35 ◦ 23 15 . Notation depends on the circumstance: some situations require angles to be given in decimal form accurate to, say, 3 decimal places; others require answers correct to the nearest tenth of one minute, for example 35 ◦ 25.7 ; and others may require an accuracy correct to the nearest second, for example 35 ◦ 23 15 . Circular measure A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius (Figure 7.5). Trigonometry and Geometry • 143 Therefore 2 π radians = 360 ◦ 1 radian = 180 ◦ π ∼ = 57.3 ◦ or 57 ◦ 17 45 1 ◦ = π 180 ∼ = 0. 01745 radians r θ s A Figure 7.5 θ is in radians Length of arc, s = r · θ Area of sector, A = 1 2 · r 2 · θ Similarly, if a wheel of 0.3 m radius turns through 4 rad in 1 s, then a point on the rim moves at a rate of 4 × 0.3 = 1.2 m/s.

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