10. In the preceding figure, what are the various triangles formed ? At what various angles are the sides of the octagon at centre inclined to any side of the original octagon ? 11. In the same figure, what angles alone occur? How many rhombuses are there in the figure ? 12. Construct a regular octagon whose side is 35 millimetres. Test the accuracy of your construction. 13. With the angular points of a regular octagon as centres, describe eight circles of equal radii, so that each touches two others of the set. 14. With respect to how many lines is a regular hexagon symmetrical? Has it central symmetry? 15. With respect to how many lines is a regular octagon symmetrical ? Has it central symmetry ? Has a regular heptagon central symmetry ? 16. In a circle of radius 37 millimetres inscribe a regular dodecagon. 17. What is the ratio of the sides of two regular hexagons, one inscribed in, and the other described about, the same circle ? 18. ABCDEF is a regular hexagon. Show that its area is twice that of the equilateral triangle ACE. 19. In a circle the angle ABC is equal to the angle BCD. How are the chords AB, CD related ? 20. An equiangular polygon inscribed in a circle has its alternate sides equal. 21. At B, a point on a circle, construct an angle ABC of 108° (the angle of a regular pentagon), the sides AB, BC not being equal. At C make BCD of 108°; at D make CDE of 108°; and so on. Shall we at length reach accurately the point A ? If so, after how many times about the circle? Has a regular pentagon been described ? Can other regular pentagons be obtained from the figure by producing lines or otherwise ? a CHAPTER XIX. Similar Triangles. 1. Two triangles are similar when the angles of one triangle are equal to the angles of the other, the sides not necessarily being equal. Thus if two triangles of different sizes have their angles 45°, 65° and 70', they are similar. In the following article a remarkable property of such triangles is reached. 2. On a base BC of 15 millimetres construct a triangle with sides AB, AC of 20 and 25 millimetres. 2 2 1 2 2 1 1 2 2 Draw two other bases B,C,, B,C, of lengths 30 nd 45 millimetres. At B, and B, make angles C,B,A1, C,B,A,, each equal to CBA; and at C, and C, make A angles B,C,A,, B,C,A,, each equal to BCA. It follows (Ch. III., 4) that the angles at A, A1, A, are equal to one another. Hence the three triangles are equiangular and similar. Now measure the lengths of the sides of the triangles A,B,C, and A,B,C,. If the constructions have been accurately made, we shall have the following numerical values : 1 2 Then calling those sides corresponding sides which are opposite to equal angles, we observe that corresponding sides about equal angles are proportional, i.e., 1 2 1 1 3. Again, construct a triangle ABC, whose base BC is 24, and sides AB and AC, 30 and 40 millimetres. Draw two other bases B, C, and B,C, of lengths 36 and 60 millimetres. At B, and B, make angles C,B,A1, C,B,A,, each equal to CBA; and at C, and C, make A angles B,C,A,, B,C,A,, each equal to BCA. It follows (Ch. III., 4) that the angles at A, A1, A, are equal to one another. Hence the three triangles are equiangular and similar. 2 2 1) 2 Now measure the lengths of the sides of the triangles A,B,C, A,B,C2. If the constructions have been accurately made, we shall have the following numerical values: 1 1 2 1 1 2 2 BC = 24 B,C, = 36 B,C, = 60 A,B,=45 A,B,= 75 A,C,= 100 And we again find that corresponding sides about equal angles are proportional, i.e., 1 1 4. The pupil may repeat this experiment with equiangular triangles, and, the constructions being accurately made, he will always reach the same conclusion as to the proportionality of the corresponding sides about equal angles. (The easiest way to secure the equality of the angles is to place with the parallel rulers B, C, parallel to BC, and then with the same rulers draw B,A, parallel to BA, and C, A, parallel to CA.) The result of these observations may be stated thus: The sides about the equal angles of equiangular triangles are proportionals; and corresponding sides, i.e., those which are opposite to equal angles, are the antecedents or consequents of the ratios. (Note: In the ratio a:b, a is called the antecedent, and b the consequent.) This is the most important proposition in Geometry: indeed, one of the inost important results of all science. Through it, in effect, all measurements are made when we cannot actually go over the distance to be measured with a rule, a surveyor's chain, or other measuring instrument. |