6. If two chords are drawn from a point in a circumference, the chord making the smaller angle with the radius to that point is the larger chord. 7. The perpendicular from the center of a circle to the side of an inscribed equilateral triangle is less than the perpendicular from the center to the side of an inscribed square. 8. If point E be taken on side AB of isosceles triangle ABC, prove EC greater than EB. (AB = AC.) 9. In triangle ABC, AB is greater than AC, and the median AE is drawn. Prove the angle AEB greater than angle AEC. 10. A method of laying out the shortest possible railroads from two towns to a common junction on the main line: Let MR be the main line, A and B the two towns. Draw A01MR, and extend it its own length to C. Draw BC inter secting MR at H, BHAH < BK + AK or any other combination. B M 1 10 1 R 11. In a given line, find a point at a given distance from a given point. 12. Find a point at a given distance from two given points. 13. Find the locus of points at a given distance from a given circumference. 14. Find the locus of the centers of circles which have a given radius and pass through a given point. 15. Find the locus of the centers of circles which pass through two given points. 16. Construct a circle which has a given radius and passes through two given points. 17. Construct a circle which has its center in a given line and passes through two given points. CHAPTER XIV MEASUREMENT OF ANGLES The basic method is: 197. A central angle has the same measure as its intercepted arc. 198. Define Measurement; Unit of Measure. 199. Define Numerical Measure. 200. Degree. What is the unit of measure for angles? 201. 202. Define Commensurable. 203. Incommensurable. *204. An inscribed angle has the same measure as one-half its intercepted arc. (Three cases.) 205. Angles inscribed in the same or equal segments of a circle are equal.. 206. An angle inscribed in a semicircle is a right angle. 207. The opposite angles of an inscribed quadrilateral are supplementary. *208. An angle between a tangent and a chord, meeting on the circumference, has the same measure as one-half the intercepted arc. *209. An angle between two chords meeting within the circle has the same measure as one-half the sum of the intercepted arcs. *210. An angle between a tangent and a secant, two tangents, or two secants, meeting without the circle, has the same measure as one-half the difference of the intercepted arcs. (Three cases.) 211. Group 197, 204, 208, 209, and 210 under one state ment. *212. PROBLEM: Erect a perpendicular at the end of a line. *213. PROBLEM: Draw a tangent to a circle from a point on the circumference. *214. PROBLEM: Draw a tangent to a circle from a point without the circle. *215. PROBLEM: Upon a given straight line describe a segment of a circle in which any inscribed angle will equal a given angle. SUMMARY 216. State the measure of each of the following angles: a. At the center of a circle. b. Between the center and the circumference. c. At the circumference, whose sides are chords or tangents. d. Outside the circle, whose sides are secants or tangents. 217. State two additional methods of proving angles equal. 218. State one additional method of proving lines perpendicular. 219. EXERCISES 1. Prove Art. 208 by drawing a radius perpendicular to the chord and a radius to the point of tangency. Prove, also, by drawing a line from the end of the chord parallel to the tangent. 2. ove Art. 209 by drawing a line from the end of one chord parallel to the other. 3. Prove Art. 210 in the same way. 4. Prove ex. 6, Art. 187, by measurement of angles. 5. Prove ex. 7, Art. 187, by measurement of angles. 6. Prove ex. 9, Art. 187, by measurement of angles. 7. If the bisector of an inscribed angle is extended to the circumference and through this point a chord is drawn parallel to one side of the angle, this chord equals the other side of the angle. 8. An exterior angle of an inscribed quadrilateral equals the opposite interior angle. 9. If two circles are tangent, a line through the point of tangency subtends equal (in degrees) arcs. 10. If two circles are tangent externally, and a line is drawn through the point of tangency and ending in the circumference, tangents at the ends of this line are parallel. E 11. If two circles are tangent externally, and two lines are drawn through the points of tangency and ending in the circumference, the chords joining the ends of these lines are parallel. F 12. If two circles intersect, and a line is drawn through each point of intersection and ending in the circumferences, the chords joining the ends of these lines are parallel. 13. In the figure, prove ▲ AOB> Z ACB; Z ACB > Z ADB; ≤ ADB > LAEB. 14. An application of ex. 13 in determining the "angle of safety" in navigation: A certain section of a bay is dangerous on account of hidden rocks and shoals. To enable sailors to keep out of this section both by day and by night, two lighthouses, A and B, are built upon the shore. Then on a map a circle is drawn through A and B, and enclosing the dangerous area. An inscribed angle ASB is measured. This is the angle of safety; and if the angle from any boat T to A and B is less than the angle of safety, the boat is in safe waters. 15. A method of finding the latitude of a point on the earth north of the equator: Let O be the center of the earth; EF the equator; P the point; PH a line to the horizon; PN a line to the north star. The measure of HPN is the latitude of P. Proof: ▲ POE is measured by PE. LHPN = Z POE. LHPN is measured by PE. As PE is the latitude of P, the measure of HPN is the latitude of P. CHAPTER XV REGULAR POLYGONS The basic method is: A regular polygon is equilateral and equiangular. (Art. 46b.) *220. If the circumference of a circle is divided into any number of equal arcs, (1) the chords to the points of division form a regular inscribed polygon; (2) the tangents at the points of division form a regular circumscribed polygon. 221. If the arcs subtended by the sides of a regular inscribed polygon are bisected, the tangents at the points of bisection form a regular circumscribed polygon of the same number of sides. 222. What is the Radius of a regular polygon? What can be said about all radii of a regular polygon? What is the Apothem of a regular polygon? What can be said about all apothems of a regular polygon? . 223. What is the Angle at the Center of a regular polygon? How many central angles are there? What can be said about all angles at the center of a regular polygon? 224. Each angle at the center equals 360° divided by what? *225. If two diameters of a circle are perpendicular, the chords joining their ends form a square. *226. PROBLEM: Inscribe a regular hexagon in a circle. *227. PROBLEM: a. Given a regular inscribed polygon, construct a regular inscribed polygon of double the number of sides. NOTE. Bisect the arcs. |