CHAPTER VIII CIRCLES 150. Circles. - The definition of a circle was given in § 6, also the definitions of radius, diameter, and arc of a circle. It was shown in § 103 that a circle is a locus of points at a given distance from a given point. A circle is divided into two arcs by any two of its points, as A and B. If these two arcs are equal, each is called a semicircle. If they are unequal, the smaller is called the minor arc and the larger is called the major arc. Unless otherwise specified, arc AB (written AB) Major B Minor Arc A Arc shall be taken to mean the minor arc terminating at A and B. An angle formed by two radii is called a central angle, as ∠AOB. An angle is said to intercept the arc cut off by its sides, and the arc is said to subtend the angle. A straight line which touches a circle at only one point is called a tangent. The point is called the point of contact. Tangent Secant Diameter Chord A straight line which intersects a circle at two points is called a secant. A line-segment whose ends are on a circle is called a chord. It is evident that a diameter is a chord which passes through the center. A chord is said to subtend the arcs into which its extremities cut the circle. Two circles which have the same center are called concentric. 151. Fundamental properties of circles. The following are fundamental properties of circles : (1) A diameter of a circle equals two times a radius. CONCENTRIC For, a diameter is composed of two radii by definition. (2) Radii of the same circle or equal circles are equal. Radii of the same circle are equal by definition of a circle. And since equal circles may be made to coincide, their radii may be made to coincide and hence are equal. (3) If the radii of two circles are equal, the circles are equal. For, by making the equal radii coincide, the circles coincide. (4) In the same circle or equal circles, equal central angles intercept equal arcs. For, when the equal angles are superposed so that they coincide, the intercepted arcs coincide. (5) In the same circle or equal circles, equal arcs subtend equal central angles. For, when the equal arcs are superposed so that they coincide, the central angles coincide. (6) In the same circle or equal circles, equal arcs are subtended by equal chords. For, when the equal arcs are superposed so that they coincide, the chords which subtend the arcs coincide. (7) In the same circle or equal circles, equal chords subtend equal arcs. For, when the equal chords are superposed so that they coincide, the subtended arcs coincide. (8) A point is at a distance from the center of a circle equal to, greater than, or less than the radius, according as it is on, outside of, or within the circle; and conversely. This follows readily from the definition of a circle and Axiom X. These fundamental properties of circles should be thoroughly thought out and understood, as well as memorized, by the student. They are very important principles, and will be used much in the subsequent work. Yet, they are so simple that more formal proofs of them than those suggested above are not demanded. EXERCISES 1. Divide a given circle into four equal arcs. SUGGESTION. Draw two perpendicular diameters. Why are the central angles equal? Why then are the arcs equal? 2. Divide a given circle into eight equal arcs. SUGGESTION. - Draw two perpendicular diameters, then draw the bisectors of the four angles at the center. 3. Explain how to divide a given circle into sixteen equal arcs. Into thirty-two equal arcs. Into sixty-four equal arcs, etc. Into how many equal arcs, in general, may a circle be divided by this method? 4. Divide a given circle into six equal arcs. SUGGESTION. Construct an equilateral triangle upon any radius as a side. How many degrees are there in the central angle? How may the construction be completed? 5. Divide a given circle into three equal arcs. 6. Explain how to divide a given circle into twelve equal arcs. Into twenty-four equal arcs, etc. Into how many equal arcs, in general, may a circle be divided by this method? 7. Construct an equilateral quadrilateral with its vertices on a given circle. SUGGESTION. — First divide the circle into four equal Then draw the chords of these arcs. Why are the chords equal? arcs. 8. Construct an equilateral hexagon with its vertices on a given circle. 9. Construct an equilateral octagon with its vertices on a given circle. 10. Construct an equilateral triangle with its vertices on a given circle. 11. These designs are made by dividing a circle into equal arcs. Study and explain their constructions. *** 12. Make an original design based upon the division of a circle into equal arcs. 13. PQ is drawn through the center O, intersecting the circle at A and B; and PR is any other secant intersecting the circle at Cand D. Prove PA < PC, and PB>PD. R D C P A 0 BQ SUGGESTIONS. - For proving that PA< 14. The arcs intercepted between a diameter and a parallel chord are equal. SUGGESTION. – Draw radii to the ends of the chord, forming central angles which intercept the arcs. The arcs may be proved equal by first proving what? The proof that the central angles are equal depends upon the proof that the triangle formed by the chord and the radii is what kind of triangle? 152. Theorem. - Through three points not in a straight line one and only one circle can be drawn. Hypothesis. A, B, and Care any three points not in a straight line. Conclusion. Through A, B, and Cone and only one circle can be drawn. Proof. 1. Draw AB, AC, and BC. Draw the perpendicular bisectors of AC and BC, and let them intersect at 0. 2. Then O is equidistant from A, B, and C. 3... a circle with center O and radius OA can be drawn through A, B, and C. § 112 § 103 4. The center of any circle passing through A, B, and C is equidistant from A, B, and C. Def. circle radius. 5. ... the center must be at O, and hence OA must be the § 105 6. ... only one circle can be drawn through A, B, and с. § 151, (3) 153. Corollary 1. Two circles can intersect in only two points. The proof is left to the student. 154. Corollary 2. - A straight line can intersect a circle in only two points. The proof is left to the student. 155. Corollary 3. - The perpendicular bisector of a chord passes through the center. The proof is left to the student. |