THEOREM XIII. 29. Similar triangles are to each other as the squares of their homologous sides. Let ABC and DEF be two similar triangles; then ABC: DEFAC2: DF2 Draw BG and EH perpendicular respectively to A C and D F ; then (22) B F BG: EHAC: DF AC: DF ACDF this multiplied by the proportion gives but = }ACXBG: DFXEH=AC:DF ACX BG is the area of ABC, and DFX EH is the area of D E F (11); therefore ABC: DEFAC2: DF2 THEOREM XIV. 30. Similar polygons can be divided into the same number of similar triangles. Let ABCDEF and GHIKLM be similar poly D G K M L F E gons; they can be divided into the same number of similar triangles. From the homologous angles A and G draw the diagonals A C, A D, A E, GI, G K, and GL; these diagonals divide the polygons as required. For, as the polygons are similar, the angle B = H, and A B : G H = BC: HI; therefore the triangles A B C and GHI are similar (23). As the triangles ABC and GHI are similar, the angle BC A = HIG; but the whole angle BCDHIK; therefore the angle ACD=GIK; and as the triangles A B C and G H I are similar BC: HI=AC: GI But Therefore ACGI CD: IK and ACD and GIK are similar (23). In like manner it can be proved that the other triangles are similar each to each. THEOREM XV. 31. The perimeters of similar polygons are to each other as the homologous sides; and the polygons as the squares of the homologous sides. AB+BC+ CD, &c. : GH+HI+IK, &c. AB: GH that is, the perimeters of ABCDEF and G HIKLM are as AB: GH. 2d. ABCDEFGHIKLM AB2: GH2 From the homologous angles A and G draw the diagonals A C, A D, A E, G I, G K, and G L; the polygons will be divided into the same number of similar triangles (30); therefore (29) ABC+ACD+ADE+AEF: GHI+GIK+GKL+ But GLM=ABC:GHI ABC: GHIAB2: GH2 Therefore the sums of the triangles, that is, the polygons themselves, are to each other as the squares of the homologous sides. 32. Definition. A Regular Polygon is one that is both equiangular and equilateral. THEOREM XVI. 33. Regular polygons of the same number of sides are similar. for the sum of their angles is the same (I. 67); and each angle is equal to this sum divided by the number of angles which is the same. The homologous sides are proportional; for as the polygons are regular, A B = BC= CD, &c., and G H =HI=IK, &c., therefore AB: GH BC: HI= CD: IK, &c. = THEOREM XVII. 34. There is a point in a regular polygon equidistant from its vertices, and also equidistant from its sides. Let ABCDEF be a regular polygon. Biscct the angles A and B by A O and BO. As the whole angles A and B are each less than the two right angles, the A sum of OAB and ABO is less than two right angles; therefore AO and BO cannot be parallel (I. 17), but will meet. B G C F E D Suppose them to meet in the point 0; then O is equidistant can be proved that OD=0E0F0A; that is, O is equidistant from the vertices of the polygon. As the triangles OA B, OBC, O CD, &c. are equal, their altitudes are equal, that is, the bases are equidistant from the vertex 0. 35. Scholium. O is called the centre, and the perpendicular OG the apothem of the polygon. 36. Corollary. In regular polygons of the same number of sides, the apothems are as the homologous sides; therefore the perimeters of regular polygons of the same number of sides are as their apothems; and the polygons as the squares of their apothems. THEOREM XVIII. 37. The area of a regular polygon is equal to half the product of its perimeter and apothem. For the area of each triangle of which the polygon is composed is equal to half the product of its base and the apothem of the polygon (11); therefore the area of the polygon is equal to half the product of the sum of the bases, that is, its perimeter and its apothem. PRACTICAL QUESTIONS. 1. What is the perimeter and the area of a rectangle 25 by 35 inches? 2. What is the area of a parallelogram whose base is 20 feet and altitude 12 feet? 3. What is the area of a triangle whose base is 14 feet and altitude 8 feet? 4. What is the square surface of a board 15 feet long, and 16 inches wide at one end and 9 inches at the other? What kind of a figure is it? 5. What integral numbers will express the sides and hypothenuse of a right triangle? 6. How far from a tower 40 feet high must the foot of a ladder 50 feet long be placed that it may exactly reach the top of the tower? 7. The foot of a ladder 67 feet long stands 40 feet from a wall; how much nearer the wall must the foot be placed that the ladder may reach 10 feet higher? 8. If a ladder 108 feet long, with its foot in the street, will reach on one side to a window 75 feet high, and on the other to a window 45 feet high, how wide is the street? 9. A has an acre of land one of whose sides is 20 rods in length; B has a piece of land of exactly similar form containing 9 acres. What is the length of the corresponding side of B's? 10. What is the distance on the floor from one corner to the opposite corner of a rectangular room 16 by 24 feet? 11. If the height of the above room is 10 feet, what is the distance from the lower corner to the opposite upper corner? 12. Find the length of the longest straight line that can be drawn in a cube whose dimensions are 12, 4, and 3. 13. What is the altitude of an equilateral triangle whose side is 12 feet? 14. If the bases of two similar triangles are respectively 100 and 10 feet, how many triangles equal to the second are equivalent to the first? 15. How many times as much paint will it take to cover a church whose steeple is 120 feet in height as to cover an exact model of the church whose steeple is 10 feet in height? 16. What is the area of a right-angled triangle whose hypothenuse is 125 feet and one of the sides 75 feet? |