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' 4 feet, find the sum of the areas of the semicircles.

(c) Find the area inclosed by the line-segments FB', B'E and the arcs FB and BE, if EB' is 13 feet.

(d) Find the areas required in (b) and (c) if A'B' = a.

D'

E

B'

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

ments.

(b) If AB = a, find BC.

DE F

(c) If AB a find the area of the square

A

xyzw.

B

(d) At what angles do the oblique lines meet the horizontal?

(e) Construct the figure by laying off the

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.

(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 lg and l

(b) What part of the area between 1, and l 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 = 12 feet, find the radii of the circles O, O', O".

(d) Find these radii if AB

=s (span of the arch).

(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.

A

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.

EL

AK Q D

(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 vertices of the triangles as centers.

(a) If AB=6 inches, find the area bounded by CQ, QP, PT, TS, SR, RC.

R

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?

(ƒ) Find the area required under (d) if AB = a.

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.

F

E

36. Prove that if the points D and F coincide in the preceding example the tangent at D is parallel to CE.

37. Two circles are tangent internally at A. Prove that all chords of the larger circle through A are divided proportionally by the smaller circle.

E

B

38. Chords are drawn through a fixed point on a circle. Find the locus of points which divide them into a fixed ratio.

39. Squares are inscribed in a circle, a semicircle, and a quadrant of the same circle. Compare their areas.

40. In a given circle two diameters are drawn at right angles to each other. On the radii thus formed as diameters semicircles are constructed. Show that the four figures thus formed are congruent. 41. Let C be any point on the diameter AB of a circle.

(a) Compare the length of the arc ADB with the sum of the lengths of the arcs AEC and CFB.

(b) Show that if AB 3 CB, then the area inclosed by the arcs BFC, CEA, ADB, is one third the area of the circle.

(c) Show that if AB = m. CB, then the area inclosed by these arcs is one mth of the area of the circle.

42. By means of arcs constructed as shown in the third figure divide the area of a circle into any given number c

of equal parts. Make the construction.

43. Two sides AB and BC of a triangle are extended their own lengths to B' and C' respectively. Compare the areas of the triangles ABC and BB'C'.

44. The three sides of a triangle ABC are extended to A', B', C' as shown in the figure. Compare the areas of the triangles ABC and A'B'C': (a) if BB' AB, CC' BC, and AA' = CA; (b) if BB' =l · AB,CC' = m · BC, and AA'=n.CA.

=

=

A

B