16. In the accompanying design for grill work: (a) Find the angles ABC, BAD, and AGC. (b) If the radius of each circle is r, find the distance AF. (c) What fraction of the area of the parallelogram GBCE lies within one circle only? (d) If the radius of each circle is r, find the distance between two horizontal lines. (e) Construct the whole figure. SUGGESTIONS. (1) Find AF and lay off points on AB. (2) Find BAD and construct it. (3) Through the points of division on AB draw lines parallel to AD. (4) From A along AD lay off segments equal to AF and through these division points construct lines parallel to AB. (5) Along DC lay off segments equal to AF. Connect points as shown in the figure. (f) From the construction of the figure does it follow that 17. Prove that the sum of three altitudes of a triangle is less than its perimeter. 18. In the accompanying design for grill work, the arcs are constructed from the vertices of the equilateral triangles as centers. (a) Prove that two arcs are tangent to each other at each vertex of a triangle. (b) Find the area bounded by the arcs AB, BC, CD, DA. (c) Find the ratio between the area in (b) and the area of the triangle DBC. 19. The character of the accompanying design for a window is obvious from the figure. Denote the radius of the large circle by R, of the semicircles by R', and of the small circles (b) Find R' and r in terms of R. (c) What fraction of the area of the large circle lies within the four small circles? (d) What fraction of the area of the large circle lies outside the four semicircles? (e) If R = 10, find the area inclosed within the four small circles. 20. In the accompanying design for a stained glass window: (a) What part of the square A'B'C'D' lies within ABCD? (b) If A'B' = of the semicircles. 4 feet, find the sum of the areas (c) Find the area inclosed by the line-segments FB', B'E and the arcs FB and BE, if EB' is 1 feet. (d) Find the areas required in (b) and (c) if A'B' = a. D' E Β' 21. The accompanying design for tile flooring consists of regular octagons and squares. The design can be constructed by drawing parallel lines as shown in the figure. (a) If a side of the octagon AB is given, find BC, DE, and EF by construction. Find the ratio of any two of these seg required points on the sides, drawing parallel lines in pencil, inking in the sides of the octagons and erasing the remainder of the lines. 22. This design for tile flooring is constructed by first making a network of squares and then drawing horizontal lines cutting off equal triangles from the squares. (a) At what angle to the base of the design are the oblique lines? (b) If each of the small squares is 6 inches on a side, find EF and HL. D K L H EFG (c) Find EF and HL if the side of a small square is a. (d) What fraction of the whole area is occupied by the black triangles? 23. Five parallel lines are drawn at uniform distances apart, as shown in the figure. (a) If these lines are 4 inches apart, find the width of the strip from which the squares are made, so that their outer vertices shall just touch, and l2, and the corresponding inner vertices shall touch l, and l (b) What part of the area between 1, and l2 would be occupied by a series of such squares arranged as shown in the figure? 24. ABC is an equilateral triangle. AO and BO bisect its base angles. OD and OE are drawn parallel to CA and CB, respectively. Show that AD = DE =EB. 25. If one base of a trapezoid is twice the other, then each diagonal divides the other into two segments which are in the ratio 1:2. DEB 26. If one base of a trapezoid is n times the other, show that each diagonal divides the other into two segments which are in the ratio 1:n. 27. Prove that if an angle of a parallelogram is bisected, and the bisector extended to meet an opposite side, an isosceles triangle is formed. Is there any exception to this proposition? Are two isosceles triangles formed in any case? 28. Prove that two circles cannot bisect each other. 29. Find the locus of all points from which a given line-segment subtends a constant angle. 30. In the figure, equilateral arches are constructed on the base AB, and on its subdivisions into halves, fourths, and eighths. (a) Show how to construct the circle O tangent to the arcs as shown in the figure. SUGGESTION. The point O is determined by drawing arcs from A and B as centers with BF as a radius (Why?). (b) Show how to complete the construction of the figure. = (c) If AB: (e) What part of the area of the arch ABC is occupied by the arch ADE? by the arch AFS? by ALK? (f) The sum of the areas of the seven circles is what part of the area of the whole arch? (g) The sum of the areas of the two equal circles O' and O"" is what part of the area of the circle O? 31. The accompanying church window design consists of the equilateral arch ABC and the six smaller equal equilateral arches. (a) If AB = 8 feet, find the area bounded by the arcs MG, GE, ED, DM. = (b) If AB 8 feet, find the area bounded by the arcs AC, CE, ED, DA. D E (c) Find the areas required under (a) and (b) if AB = a. 32. In the figure, ABC is an equilateral arch. D, E, and F, the middle points of the sides of the triangle ABC, are centers of the arcs AE, KL, BF, and SR; CF and MR; EC and LM respectively. (a) Prove that the arc with center D and radius DA passes through the point E. (b) Prove that arcs with centers D and F, and tangent to the segment AC, meet on the segment BE. (c) If AB = a, find KS. (d) Can we find the area bounded by the segment AB and the arcs BF, FC, CE, and EA when AB is given? If so, find this area when AB = a. (e) Can we find the area bounded by KS and the arcs SR, RM, ML, LK, when AB is given? If so, find this area when AB = 6 feet. 33. Prove that the altitude of an equilateral triangle is three times the radius of its inscribed circle. 34. The accompanying grill design is based on a network of congruent equilateral triangles. Arcs are constructed with ver tices of the triangles as centers. (a) If AB=6 inches, find the area bounded by CQ, QP, PT, TS, SR, RC. M P N H R C D G A B E (b) Has the figure consisting of these arcs a center of symmetry? How many axes of symmetry has it? (c) Find the area required under (a) if AB = a. (d) If AB = 4 inches, find the area bounded by CD, DE, EF, FG, GH, HK, etc. (e) Has the figure consisting of these arcs a center of symmetry? How many axes of symmetry has it? (f) Find the area required under (d) if AB = α. 35. Two circles intersect in the points A and B. Through A a line is drawn, meeting the two circles in C and D respectively, and through B one is drawn meeting the circles in E and F respectively. Prove that CE and DF are parallel. D F B B E |