SECTION III. OF THE CIRCLE, AND LINES OF THE CIRCLE. B 1. What is a circle? What is a circumference? A circle is a plane figure, bounded by a curve line, all the points of which are equally distant from a certain point within, called the centre. The curve line is called the circumference. Thus, the space enclosed by the curve ABD is called a circle: the curve ABD is the circumference, and the point C, the centre. 2. What is the radius of a circle? Are all radii equal? Any line, as CA, drawn from the cen tre C to the circumference, is called a radius, and two or more such lines, are radii. All the radii of a circle are equal to each other. 3. What is the diameter of a circle? How does it divide the circumference? The diameter of a circle is any line, as AD, passing through the centre and terminating in the circumference. Every diameter of a circle divides it into two equal parts, called semicircles, or half circles. 4. What is an arc? An arc is any portion of the circumference. Thus, AEB is an arc. 5. What is a chord? A chord of a circle, is a line drawn within a circle, and terminating in the circumference, but not passing through the centre. Thus, AB is a chord. A chord divides the circle into two unequal parts. 6. What is a segment? A segment of a circle, is a part cut off by a chord. Thus, AEB is a seg- A ment. The part AOB, is also a segment, although the term is generally applied to the part which is less than a semicircle. 7. What is a sector? A sector of a circle, is any part of a circle bounded by two radii and the arc included between them. Thus, ACB is a sector. B 8. What is an angle at the centre? An angle at the centre, is one whose vertex is at the centre of the circle. Thus, BCE, or ECD, is an angle at B the centre. 9. What is an angle at the circumference? An angle at the circumference, is one whose angular point is in the circumference. Thus, BAC, or BOC, is an angle at the circumference. 10. What is an angle in a seg ment? An angle in a segment, is formed by two lines drawn from any point of the segment to the two extremities of the arc. Thus, ABE is an angle in a seg ment. 11. What is a secant line? A secant line, is one which meets the circumference in two points, and lies partly within and partly without. Thus, AB is a secant line. 12. What is a tangent line?—What position has it unth the radius passing through the point of contact? A tangent is a line which has but one point in com Thus, mon with the circumference. Thus, CM is E C D 13. When is a figure said to be inscribed in a circle ?— What is said of the circle? A figure is said to be inscribed in a circle when all the angular points of the figure are in the circumference. The circle is then said to circumscribe the figure. Thus, the triangle ABC is inscribed in the circle, and the circle circumscribes the triangle. A B 14. When is a figure said to be circumscribed about a circle? A figure is said to be circumscribed about a circle, when all the sides of the figure touch the circumference. The circle is then said to be inscribed in the figure. 15. How is an angle at the centre of a circle measured? An angle at the centre of a circle is measured by the arc contained by the sides of the angle. This arc is said to subtend the angle. Thus, the angle ACB is measured by the degrees in the arc AEB, and is subtended by the arc AEB. 16. What measures an angle at the circumference? An angle at the circumference of a circle, is measured by half the arc which subtends it. Thus, the angle BAD is measured by half the arc BD. Hence, it follows, that when an angle at the centre and an angle at the circumference stand on the same arc BD, the angle at the centre will be double the angle at the circumference. 17. What is the sum of the three angles of any triangle equal to? To 180 degrees, since they will be measured by one half of the entire circumference. B 18. What is an angle in a semicircle equal to? An angle inscribed in a semicircle, is a right angle. Thus, if AB be the diameter of a circle, then will the angle ACB be equal to 90 degrees. This angle is measured by one half the semi-circumference, that is, by one half of 180°, or by 90°. 19. Are the arcs intercepted by parallel chords equal, or unequal? Two parallel chords intercept equal arcs. That is, if the chords AB and CD are parallel, the arcs AC and DB, which they intercept, will be equal. A B |