example, in § 147 and 156, but the above process gives 3.1414 times the square of the radius instead of times, and the error is less than the five thousandth part of the square of the radius. Prove that- Questions for Examination. 1. Two regular polygons are equal when a side of the one is equal to a side of the other and their interior angles are equal. 2. When they have the same number of sides, and are inscribed in equal circles. 3. Upon a given straight line A B constructI. A hexagon. dodecagon. 2. An octagon. 3. A decagon. 4. A 4. Construct a polygon equal to a given polygon, 5. Construct a polygon symmetrically equal to a given polygon. 6. Divide a plot of ground into six equal portions by paths leading to a house. 7. In a given quadrilateral inscribe a square which shall be half the quadrilateral. 8. To construct a polygon similar to another. 9. Similar polygons are to one another as the squares of their homologous sides. 10. Construct a regular hexagon which shall be equal to the sum of two given similar hexagons. II. Construct a semicircle which shall be equal to the sum of three given semicircles. 12. Prove that circles are similar figures, and their circumferences are to one another as their radii. 13. Show how to find the ratio of the circumference of a circle to the diameter. Prove that this ratio is between 3 and 4. 14. Draw a straight line nearly equal to a given circumference. 15. Find the area of a given regular polygon. 16. Find the area of a circle when the radius is given. 17. Describe a square equivalent to a given circle. Theorems and Problems. 1. On a given straight line construct the following regular polygons by describing an arc with one extremity and making a suitable division of the quadrant, I. Pentagon. 2. Hexagon. 3. Heptagon. 4. Octagon. 5. Nonagon. S 2. On a given straight line construct the following regular polygons by making adjacent angles at each extremity of this line equal to half the centre angle of the polygon, and producing the sides so that they are bisected by the perpendicular at the middle point of the given line. Polygons of 3, 5, 8, 9, 10, and 12 sides. 3. In a given triangle describe a parallelogram similar to a given parallelogram, by the following construction. Construct a parallelogram on the base similar to the given parallelogram; draw a line through the apex parallel to sides of this parallelogram; from the point where this line meets the base, draw lines to the opposite angles of the parallelogram. 4. Solve the last problem when the parallelogram is a square. 5. Construct a polygon equal to one polygon and similar to another. 6. Prove that if squares be described on the sides of a rectangle, and their diagonals be made the sides of another rectangle, the latter will be half the former. 7. Prove that the area of a regular octagon inscribed in a circle is equal to that of a rectangle which has for its sides the sides of the inscribed and circumscribed squares. 8. The inscribed hexagon is three-fourths of the circumscribed hexagon. 9. In different circles the sectors whose angles are inversely as the squares of the radii are equal. 10. Construct a triangle the area and angles of which are given. II. Divide a circle by radii into parts proportional to their number. 12 Inscribe a square in a sector of a circle, so that the angular points shall be one on each radius, and the other two in the circumference. 13. Inscribe a square in a given regular pentagon. 14. The square inscribed in a circle is to the square inscribed in the semicircle :: 5 : 2. 15. The square inscribed in a semicircle is to the square inscribed in a quadrant of the same circle :: 8: 5. 16. It two diagonals of a regular pentagon be drawn to cut one another, the greater segments will be equal to the side of the pentagon, and the diagonals will cut one another in extreme and mean ratio. 17. A regular hexagon inscribed in a circle is a mean proportional between an inscribed and circumscribed equilateral triangle. 18. The area of the inscribed pentagon is to the area of the circumscribing pentagon, as the square of the radius of the circle inscribed within the greater pentagon is to the square of the radius of the circle circumscribing it. Arithmetical Questions. 1. What is the area of a pentagon whose side measures 185 in. and the radius of the inscribed circle 127 in. Ans. 584916 sq. in. 2. Find the area of a heptagon whose side measures 2 ft. 11 in., and the perpendicular from the centre 3 ft. 3. If the diameter of a well be 3 ft. 9 in., ference? 22 Ans. 30 sq. ft. what is its circumAns. II ft. 9 in. Note.-Use as an approximation to π. 7 4. The diameter of a circular plantation is 100 yards; what did it cost fencing round, at Is. 3d. a yard? Ans. £19 12s. 10 d. 5. If the radius of a circle be 10, what are the sides of a regular inscribed pentagon, hexagon, octagon, and decagon ? Ans. 11756; 10; 7·654; 6·18. 6. If the centre of a circle whose diameter is 20, be in the circumference of another circle whose diameter is 40, what will be the areas of the three included spaces? Ans. 173 852; 140°308; and 1116.332. 7. The side of a pentagon is 30 ft., and the radius of the inscribed circle 20 6457 ft.; find its area. Ans. 172 04 sq. yds. 8. The radius of a circle circumscribing a hexagon is 14 in.; find the radius of the inscribed circle and the area of the hexagon. Ans. 1212435 and 509.2229 sq. in. 9. The side of an octagon is 5 ft.; find the radius of the inscribed circle, and the area. Ans. 6035 ft., and 120.7106 sq. ft. 10. If the diameter of a circle be 10, find the side of a square equal to it in area, Ans. 8.8623. II. Find the length of the side of a square equal in area to a Ans. 5.6419. circle whose circumference is 20 ft. 12. The diameter of the sun is 883320 miles; what is its circumference? Ans. 2,774,724 miles. 13. If the equatorial diameter be 79249 miles, what is the length of a degree of the equator? Ans. 69.156 miles. 14. Find the radius of the inscribed and circumscribed circles, and the areas of the following regular polygons : (1) Triangle; side = 10. = 5. Ans. 5 ft. 15. What is the diameter of a stone column whose circum. ference measures 16 ft. 6 in. ? 16. If the circumference of the earth be 25,000 miles, what is its diameter, supposing it to be a sphere? Ans. 7954 miles. 17. The diameters of two concentric circles being 15 and 9, required the area of the ring contained between their circumferences. Ans. 113.0976. 18. What is the area of the ring, the two diameters being 10 and 20? Ans. 235.62. 19. A carpenter is to put a stone curb to a round well, at 8d. per foot square; the breadth of the curb is to be 7 in., and the diameter of the well 31 ft. ; what will be the expense? Ans. 5s. 24d. 20. Seven men bought a grinding-stone of 60 in. diameter, each paying part of the expense; what part of the diameter did the first and last grind down? Ans. The ist, 4'4508; 7th, 22·6778 in. 21. Find the area of a circle whose diameter is 7. Ans. 38. 22. How many sq. yds. are in a circle whose diameter is 3 it.? Ans. 1'069. 23. Find the area of a circle whose circumference is 12 ft. Ans. 11'4595. 24. Find the area of the sector whose radius is 63 ft. and contained angle 30° 40'. Ans. 1062.6 sq. ft. 25. Find the area of the sector whose radius is 5 ft. 10 in. and contained angle 128° 6'. 26. What must be the length of a chord a circle including I acre of ground. Ans. 37.88 sq. ft. which will strike out Ans. 39 25068 yds. whose base falls on the 27. The area of an equilateral triangle, diameter, and its vertex in the middle of the arc of a semicircle, is equal to 25, what is the diameter of the circle? Ans. 13.16074. 28. In an isoceles triangle, two circles are inscribed touching each other and the sides of the triangle; the diameters of the circles are 9 and 25; required the sides of the triangle. Ans. 44 27083, 44°27083, and 41*66666. 29. The radius of a circle is 10 ft.; find the sum of the areas of the segments cut off by the sides of a regular inscribed hexagon. Ans. 14 159 sq. ft. CHAPTER XIV, Miscellaneous Problems 353. The proposition proved in page 72 is frequently of use in solving problems on the intersection of lines: —“A straight line joining the middle points of two sides of a triangle is parallel to the third and equal to half the third." We will here give a few problems in which this fact is applied. 354. (1) The three middle lines of a triangle meet in a point which divides each middle line into two parts, one of which is double of the other. E m n B Let O (fig. 375) be the point of intersection of two middle lines DC and EB. Draw the line m n connecting the middle points m and n of the lines BO, CO. Also draw the line DE. Because D and E are middle points of the sides of the triangle ABC, DE is parallel to BC, and DE= BC. Because m and n are middle points of the sides of the triangle OBC, mn is parallel to BC, and mn= BC, therefore DE and mn are parallel and equal. Consequently Dn and m E are diagonals of a parallelogram, and therefore bisect one another; hence mO = EO, and nO = DO, therefore EO BE, and DO DC. Fig. 375. If now we take a similar construction with DC and the third middle line AF, we shall be able to prove that AF cuts DC in a point one-third of its length from D; therefore the third middle line also passes through the point O. Hence the three middle lines intersect in the same point which divides each into two parts, one of which is double of the other. |