20. ABC is an equilateral gothic arch. (See page 109.) Find the area inclosed by the segment AB and the arcs AC and BC, if AB = 3 feet. SUGGESTION. Find the area of the sector with center A, arc BC, and radii AB and AC, and add to this the area of the circle-segment whose chord is AC. 21. ABC is an equilateral triangle and A, B, and C are the centers of the arcs. Show that the area of the figure formed by the arcs is three times the area of one of the sectors minus twice the area of the AABC. 22. In the figure ABC is an equilateral triangle and D, E, and F are the middle points of its sides. Arcs are constructed as shown. (a) If AB=6 feet, find the area inclosed by CE, EF, FC. (b) Find the area inclosed by AE, EC, and CA. (c) Find the area inclosed by DF, FE, and ED. A P D B From the Union Park Church, Chicago. SUGGESTION. What kind of a triangle is HOK? In the following let AB 8 feet. = (c) Find the area bounded by the arcs DF, FE, and ED. (d) Find the area of the rectangle ABKH. (e) Find the area bounded by AH and the arcs HE and EA. (f) Find the area bounded by the upper semicircle AB and the arcs AC and BC. (g) Find all the areas required in (c) (f) if AB = a. 24. In the figure, ABCDEF is a regular hexagon. B is the center of the arc ALC, D the center of CHE, and F the center of EKA. = 16 inches, find If AB (a) The circumference and area of the circle, (b) The area bounded by the arc ALC and the segments AB and BC, (c) The area bounded by KA, AL and LK, K (d) The area bounded by ALC, CHE and ERA. (e) Find the areas required in (a)-(d) if AB = a inches. 25. ABC..... is a regular octagon. Arcs are constructed with the vertices as centers as in the figure. (a) If AB = 10 inches, find the area inclosed by the whole figure. Also if AB = a. 26. ABCDEFGH is a regular octagon. Semicircles are constructed with the sides as diameters. (a) If AB 10 inches, find the area of the whole figure. Also if AB = a. (b) Complete the drawing in the outline figure to make the steel ceiling pattern here shown. 27. In the figure ACB, AFD, DEB, and ECF are semicircles. EF is tangent to two semicircles. (a) Prove that the semicircles AFD, FCE, and DEB are equal, D being given the middle point of AB. If AB 48 inches, find: = (b) The area bounded by AC, FC and FA, (c) The area bounded by FD and DE, and the line-segment FE, A IF H H HA (d) The area bounded by AF, FCE, EB, and the line-segment AB. (e) Find the areas required in (b)-(d) if AB = a. 28. (a) If a side of the regular hexagon in the figure is a, find the area inclosed by the arcs (including the area of the hexagon). (b) Show that a circle may be circumscribed about the whole figure. (c) Find the area inside this circle and outside the figure in (a). MISCELLANEOUS PROBLEMS AND APPLICATIONS. 1. Given a rhombus, two of whose angles are 60°, to divide it into a regular hexagon and two equilateral tri angles. See Ex. 7, page 78. 2. What fraction of the accompanying design for tile flooring is made up of the black tiles? Show how to construct this design by marking off points along the border and drawing parallel lines. 3. In the accompanying design for a parquet border two strips of wood appear to be intertwined. (a) If the border is 8 inches wide and 3 feet 4 inches long, find the area of one of these strips, including the part which appears to be obscured by the other strip. (b) If the figure consists of squares, find the angle at which the strips meet the sides. Use the table on page 139. (c) If the width of the border is a and its length b, find the combined area of these strips. Compare the total area of the obscured part of these strips with the sum of the areas of the small triangles along the edge of the border. 4. A solid board fence 5 feet in vertical height running due north and south is to be built across a valley, connecting two points of the same elevation. Find the number of square feet in the fence if the horizontal distance is 80 rods. 80 rods 5. Are the data given in the preceding problem sufficient to solve it if the fence required is to be an ordinary fourboard fence, each board 6 inches wide? 6. Two circles are tangent internally at a point A. Chords AB and AC of the larger circle are drawn meeting the smaller circle in D and E respectively. Prove that BC and DE are parallel. E B A 7. Two circles, radii r and r', are tangent internally. Find the length of a chord of the larger circle tangent to the smaller if: (a) The chord is parallel to the line of centers. (b) The chord is perpendicular to the line of centers. (c) Meets the larger circle at the same point as the line of centers. 8. Given a straight line and two points A and B on the same side of it. Find a point C on the line such that the sum of the segments AC and BC shall be the least possible. SOLUTION. In the figure let B' be symmetric to B with respect to the line. Draw AB' meeting the line in C. Then C is the required point. For let C' be any other Then point on the line. AC+ C'B> AC + CB. The proof depends upon § 128 and Ax. III, § 61. Give it in full detail. 9. If in the figure preceding, AD is perpendicular to the line, prove that ▲ ADC ~ ▲ CBE and hence E = A B AD DC = C, find CD and CE. A 10. If in Ex. 8, DE = a, AD = b, BE 11. Two towns, A and B, are 10 and 6 miles respectively from a river and A is 12 miles farther up the river than B. A pumping station is to be built which shall serve both towns. Where must it be located so that the total length of water main to the two towns shall be the least possible? 6 Miles B 12 Miles 10 Miles 12. Two factories are situated on the same side of a railway at different distances from it. A spur is to be built to each factory and these are to join the railway at the same point. State just what measurements must be made and how to locate the point where these spurs should join the main line in order to permit the shortest length of road to be built. 13. Two equal circles of radius r intersect so that their common chord is equal to r. Find the area of the figure which lies within both circles. 14. In the accompanying design for oak and mahogany parquet flooring the large squares are 6 inches and the small black ones 2 inches on a side. What fraction of the whole is the mahogany (the black squares)? 15. Construct circles on the three sides of a right triangle as diameters. Compare the area of the circle constructed on the hypotenuse with the sum of the areas of the other two. Prove. 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. |