10. A baseball diamond is a square 90 ft. on a side. What is the distance from first to third base? 11. (a) In machine shops, "stock" comes in rods of different sizes with circular ends. All square rods must be cut from such round stock. (b) What must be the diameter of the round stock used to cut a square rod 1" on a side? A 2" rod? A 21" rod? 12. A window is 36 feet above ground. How far out from the foot of the wall must a 45 ft. ladder be placed to just reach the window? 13. The side of one square is 32 in. and that of another is 17 in. What is the side of a square equal to the sum of these squares? 14. If the sum of two squares is 26 square inches and one of the squares is 14 square inches, what is the side of the other square? 15. A shelf 1 foot wide is 5 ft. from the floor. The foot of a ladder is placed 5 ft. from the wall. How long must the ladder be to reach the shelf? 16. A ladder 42 ft. long can be so placed that it will reach a window 31 ft. above the ground on one side of the street, and by tipping it back without moving its foot, it will reach a window 19 ft. above the ground on the other side. Find the width of the street. = 17. In a right triangle, a 13.6", b = 16.9". Find c. 18. An equilateral triangle is 20 inches on a side. Find its altitude. 19. The base of an isosceles triangle is 136 ft. Its altitude is 60 ft. How long is its side? What is its perimeter? 20. (a) Draw an equilateral triangle ABC and bisect ZC by the line CD. (b) How does CD cut AB? (c) What kind of an angle is / ADC? (d) Cut along the line CD and describe the two parts. (e) Measure the of AADC. (f) How does the shortest side compare with the hypotenuse? (g) Does this relation hold true between the shortest side and the hypotenuse of every 30°-60° right A? 21. How do the two perpendicular sides of a 45°-45° right triangle compare? 22. If the vertex angle of an isosceles triangle is 80°, what is the size of each angle at the base? 23. Construct an isosceles triangle with a base 21", whose vertex angle is 70°. 24. (a) A barn is 60 ft. long, 40 ft. wide, and 30 ft. high. The slop- 24/ ing edge of the roof CHAPTER TEN SIMILAR FIGURES A. SIMILAR RECTANGLES 1. Draw four rectangles as follows: 2. What is the ratio of the height to the base in each of the four ? 3. Which ones have the same ratio? 4. Which ones may be considered small maps of another? 5. Draw diagonals in Figs. A and C. Measure them and find their ratio. 6. Cut out Fig. A and place on Fig. C so that the centers are together and the diagonals take the same direction. What position do the sides of Fig. A take compared with those of Fig. C? 7. Draw a fifth rectangle E, 1" by 3". Draw the diagonals and place Figs. A, C, and E together. 8. Try to place Fig. A with Fig. B or Fig. D. 9. Which of these are alike in shape? Figures that are alike in shape are called similar figures. B. SIMILAR PARALLELOGRAMS 1. (a) Draw four parallelograms: 50° B. Sides 1" and 2′′ and included 80° C. Sides 1" and 2′′ and included ▲ 50° A B 2" D (b) What is the ratio of the side to the base in each? (c) Is Fig. B equal to Fig. C? Why or why not? (d) Draw the diagonals, cut out Figs. A and B and try to fit them on each other and the others. (e) Which ones are similar? (f) Compare the (g) In similar figures, in the pairs of similar figures. (1) What must be true about their respective 4, taken in pairs? (2) What must be true about the ratio of their respective sides? 2. (a) Draw a parallelogram similar to ABCD that is 3 times as large. (b) We sometimes letter two similar figures alike except that we put a little accent mark (') beside the letters of the second figure. A' is read A prime. The A'B'C'D' is read the parallelogram A prime, B prime, C prime, D prime. (c) A' is made equal to Z A. LB' is made equal to Z B. (d) We call A and A' corresponding angles in similar figures, or homologous angles. (e) What side corresponds to AB? to BC? to CD? (f) What corresponds to Z D? (g) What is the ratio of AB to A'B'? of BC to B'C'? (h) In similar figures what must be true of all corresponding or homologous angles? (i) The word similar is used so much that it is convenient to have a symbol for it. The double curve (~) means is (or are) similar to. ABCD ~ A'B'C'D' means the parallelo |