B B 11. In the adjoining double quatrefoil the triangle ABC is equilateral. Find the area and the perimeter of the pattern, making no allowance for the overlapping of the two strips. The arcs of each strip are concentric, their centers being the vertices of the square. (Observe that the altitude of A ABC is known. Then the length of AB can be found.) C C 12. On a network of equal squares of side a construct the vaselike figure shown in the diagram. The center of each arc is at the center of a square. Find the perimeter and the area of the figure. 13. The centers of the four arcs in the adjoining figure are the vertices of the square. Prove that ABCD is a square, and find its area, each side of the larger square being 2 a. Find also the area of the quatrefoil whose vertices are A, B, C, and D. Suggestion. Observe that AD subtends an arc of 30°. MULTIFOILS a B C Any regular polygon may be used for the construction of figures similar to those suggested for the square. 14. A rose window of six lobes is to be placed in a circular opening 14 ft. in diameter. It is to have the form shown in the diagram. Determine the radius of the interior hexagon; the perimeter of the hexafoil; the area of the hexafoil. (How does a radius of one of the smallest circles compare in length with a radius of the largest circle?) 15. About the vertices of a regular inscribed hexagon as centers, and with radii equal to the radius of the given circle, describe arcs within the circle. If the radius of the circle is a, find the perimeter and the area of the resulting hexafoil. 14 ARCHES Arches are of great importance in architecture. The most common forms are semicircular, segmental, and pointed. In the diagrams AB is the width or span of the arch, while CD is its height. Owing to the symmetry of these arches, numerical computations involving these forms are greatly simplified. In the following problems, unless the context suggests a different meaning, let s = span = AB, h = height = CD, p = perimeter of the arch = length of arc AD + are BD, A = area of the arch = area inclosed by p and s. 16. Draw a semicircular arch. Let s = a. Find h, p, A. 17. The Arch of Triumph in Paris is 162 ft. in total height, 147 ft. in width, and 73 ft. in depth. The crown of its vast arch, lar, is 96 ft. from The is 48 ft. Find the perimeter of the arch; the area of the interior passage of the arch; the entire exterior surface of the structure, disregarding cornices and other projections. 18. In the adjoining design of a window let AB = 4 a. Find the length of each arc and the area of each part of the figure. 19. If AB = 4a in the following design, find the radius FD of the upper circle, and then determine the area lying between the circles. Analysis. Let FD = x. D H K a A E C F B D 20. In the adjoining window design let AB = 6a. O and O' are the vertices of equilateral triangles constructed on CE and CF respectively. Find the area of circles O and O', and the total area of the crescents between the circles. 21. A segmental arch over a door subtends a central angle of 60°. If the door is 10 ft. high and 4 ft. wide, find h, p, and A. 22. The equilateral Gothic arch has already been defined (Ex. 13, p. 190). Find the area of an equilateral Gothic arch if s = 12. 23. The horseshoe arch is used extensively in Moorish architecture. With AC AC(=) as a radius and A as a center, construct a quadrant CF. Trisect 60% 10 Then O is H the center of the arch (radius OA). If s = 2a, E F find h, p, and A. Suggestion. In the rt. AACH, ∠A = 30° and AC = a. Find CH and prove OA = НА. 24. The segmental pointed arch has its centers below the span. In the figure CM = OA. Cand D are the centers of the arcs BE and AE respectively. If s = 4 a, and OM = b, find ME. Suggestion. CE = CB. Observe that CB is the A hypotenuse of a right triangle with legs b and 3 a. 25. A four-centered arch may be constructed as follows: Divide the span into four equal parts. On CD construct a rectangle, making CF = AB. Draw the lines FH and EK. Cand Dare the centers of the small arcs, and E and F of the larger arcs. If s = 4 a, find EK. 26. The Brescia arch (Turkish) is constructed by dividing s into 8 equal parts. On AC and BD, each containing three of these parts, equilateral triangles are constructed. The arcs AF and EB have their centers at C and D respectively. At Fand E draw tangents to these arcs meeting in H. If s = 8 a, find h, p, and A. (Draw FE.) 27. The figure represents a Persian arch. Triangles ABC and DEF are congruent and equilateral. The centers of the upper arcs MC and NC are respectively D and F, while the lower arcs are drawn with the center E. Prove that the area of the arch equals the area of triangle ABC. 28. In the adjoining modification of the Persian arch the four arcs are equal quadrants. If s = 4 a, prove that the area of the arch is 4 a2. R A A F F E CMD B C D B E H E CD B C D F 29. Draw the outline of window tracery in the figure, using the given dimensions. The arch is equilateral, and all the arcs are of equal radius. Find the height of each of the pointed arches within the equilateral arch. N DB F 30. In the adjoining diagram, representing a Gothic window, AB = BC = CD = DE. The arches are all equilateral. The point O is the intersection of arcs having the radii AD and EB (Ex. 14, p. 190). The construction of the trefoil is explained in Ex. 3. If AB = a, find the area of each part of the figure, neglecting the tracery. A 31. The drop-pointed arch is formed by two arcs whose radii are less than the span. In the figure, AB is trisected at C and D. The arcs are constructed with Cand Das centers, and radii CB and DA respectively. If s = 6 a, find h. Prove that h = .645 × s approximately. 32. The Early English or lancet arch is formed by two arcs whose radii are greater than the span. In the figure the span AB is bisected at C. Construct the squares CE and CF. Make CH = CE, and CK = CF. The arcs are constructed with H and K as centers, and with radii HB and KA respectively. If AB = 2 a, find h. Prove that h = .979 × s approximately. MOLDINGS AND SCROLLS HA The construction of moldings and scrolls is based very largely on the principles of circles that are tangent internally or externally (see Book II, Proposition XIV). These designs are of importance in a number of industries and trades. 33. Draw the scotia molding shown in the margin. The curve is made up of two quadrants of 1 in. and in. radius respectively. How long is the curve? 34. Draw the cyma recta molding shown in the margin, using the given dimensions. The curve is composed of two quadrants of equal radii, tangent to each other and to the lines AB and CD respectively. How long is the curve? 14 14 2 A B CD |