b. Given a regular circumscribed polygon, construct a regular circumscribed polygon of double the number of sides. 228. Define Perimeter.

229. For discussion:

a. The circumference of a circle lies between the perimeters of the inscribed and circumscribed regular polygons. b. The area of a circle lies between the areas of the inscribed and circumscribed regular polygons.

c. Of two regular polygons inscribed in a circle, the one with the greater number of sides is more nearly equal to the circle, and its perimeter is more nearly equal to the circumference.

d. Of two regular polygons circumscribed about a circle, the one with the greater number of sides is more nearly equal to the circle, and its perimeter is more nearly equal to the circumference.

e. If the number of sides of a regular inscribed or circumscribed polygon is indefinitely increased:

1. The perimeter of the polygon is said to approach the circumference as its limit.

2. The area of the polygon is said to approach the area of the circle as its limit.

230. PROBLEM: If the radius of a circle is 1, the circumference is > 6, and <8.

SUMMARY

231. State two methods of proving polygons regular.

232.

EXERCISES

1. An equilateral polygon inscribed in a circle is regular.

2. An equiangular polygon circumscribed about a circle is regular.

3. Circumscribe a square about a circle.

4. Circumscribe a regular hexagon about a circle.

5. Inscribe an equilateral triangle; circumscribe an equilateral triangle.

6. Inscribe and circumscribe regular polygons of eight sides; sixteen sides; twelve sides.

9. The central angles of two regular polygons, of the same number of sides, are equal.

10. Find the number of degrees in the angle at the center of a regular octagon; of a regular polygon of ninety-six sides.

11. Find the number of degrees in an interior angle of a regular dodecagon; of a regular polygon of sixty-four sides.

12. How many sides has a regular polygon, whose angle at the center is 30°?

13. The radius drawn to any vertex of a regular polygon bisects the angle at the vertex.

14. The apothem of an inscribed equilateral triangle equals one-half the radius.

15. On a given straight line as a side, construct a square; regular octagon; regular hexagon; regular dodecagon.

233.

CONSTRUCTION PROBLEMS

Construct one and only one triangle, having given:

1. Two sides and the included angle.

(Art. 81.)

2. Two angles and the included side. 3. Three sides. (Art. 66.)

4. Two angles and an opposite side.

(Art. 82.)

(Find the third angle.)

5. Two sides and an opposite angle, provided that the opposite side is greater than the other given side.

Which of the previous five cases is each of the following cases of constructing one and only one right triangle, having given:

Let the base angles of an isosceles triangle be B and C, the opposite sides b and c respectively, the base a, and vertex angle A:

Construct an isosceles triangle, having given:

234. State four cases of constructing one and only one triangle.

235. State the four steps in the analysis of construction problems.

NOTE. What is the custom in naming the lines and angles of a triangle?

1. Define Circle; Circumference; Radius; Arc.

2. Define Secant; Chord; Diameter.

3.

Define Tangent; Central Angle; Inscribed Angle.

4. Define Segment; Sector; Inscribed Circle; Circumscribed Circle.

5. When are two circles equal?

6. Are two circles congruent if they are equal?

7. State two methods of proving a line tangent to a circle.

8. There are now eight methods of proving lines perpendicular. Name them.

9. How many points determine a straight line?

10. How many points determine a circle?

11. Can a circle be drawn through three points in a straight line? 12. Equality of arcs, being a new subject, is first proved by what fundamental principle of equality?

13. Chords, being straight lines, are proved equal by what method? 14. State four methods of proving arcs equal.

15. There are now ten methods of proving lines equal. Name most of them.

16. Inequality of arcs, being a new subject, depends upon what axiom?

17. Inequality of chords will be proved by the methods of what class?

18. State two methods of proving arcs unequal.

19. There are now nine methods of proving lines unequal. Name most of them.

20. Define Measurement; Numerical Measure; Unit of Measure. 21. What is a Degree?

22. What is a unit of measure for angles?

23. What is meant by Commensurable and Incommensurable quantities?

24. What is the basic theorem for measurement of angles?

25. What is the most useful theorem?

26. What can be said of the opposite angles of an inscribed quadrilateral?

27. Group 197, 204, 208, 209, and 210 under one statement. 28. An angle at the center of a circle is measured by what?

29. An angle between the center and the circumference is measured by what?

30. An angle at the circumference whose sides are chords or tangents is measured by what?

31. An angle outside a circle whose sides are secants or tangents is measured by what?

32. State two new methods of proving angles equal.

33. Define Regular Polygon; its Radius; its Apothem; its Angle at the center.

34. Each angle at the center of a regular polygon of n sides equals what?

35. Each interior angle equals what?

36. The side of a regular inscribed hexagon equals what?

37. A regular triangle is called what? A regular quadrilateral? 38.

Define Perimeter.

39. The circumference of a circle lies between what?

40. If the number of sides of a regular inscribed or circumscribed polygon is indefinitely increased, how does the polygon change? 41. State two methods of proving polygons regular.

42. What is a method of inscribing a regular polygon in a circle? 43. What is meant by: A central angle is measured by its intercepted arc?

44. The apothem of a regular circumscribed hexagon equals what? 45. The side of a circumcribed square equals what?

46. The diagonal of an inscribed square equals what?

47. State four steps in the analysis of construction problems. 48. What is the custom in naming the sides and angles of a triangle? 49. How is distance measured from a point to a straight line? From a point to a circle?

50. What can be said about equal chords in a circle?

MISCELLANEOUS EXERCISES

1. The perpendicular bisectors of the sides of an inscribed polygon meet at a common point.

2. The perpendiculars to the sides of circumscribed polygon at the points of tangency meet at a common point.

3. The bisector of the angle between two tangents to a circle passes through the center.

4. The bisectors of the angles of a circumscribed polygon meet at a common point.

5. An inscribed trapezoid is isosceles.

6. A chord is parallel to the tangent at the middle point of the subtended arc.

7. A circumscribed parallelogram is equilateral.

8. If two circles are concentric and a common secant is drawn, the parts of the secant between the circumferences are equal.

9. If two equal secants are drawn from a point to a circle, the parts within the circle are equal.

10. If two equal secants are drawn from a point to a circle, they are equally distant from the center.

11. The radius of the circle inscribed in an equilateral triangle is one-half the radius of the circumscribed circle.

12. If chord BA of a circle is produced to C making AC equal the radius, and COE is drawn through the center, ▲ BOE = 34 C.

13. If perpendiculars are drawn from the ends of a diameter to a tangent, the sum of the perpendiculars will equal the diameter.

14. If two circles are tangent externally, the common interior tangent bisects their common exterior tangent.

15. The common interior tangents of two circles are equal.