Similar Surfaces 729. Figures that are of exactly the same shape though they differ in size are called similar figures. All circles are similar; also all squares, and all regular polygons having the same number of sides. Two maps of the same country drawn to different scales are similar figures. In order that polygons may be similar, for every angle of the one there must be a corresponding equal angle of the other and the sides about the equal angles must be proportional. 730. 1. Triangles ABC, DEF, and GHJ are similar. do the sides of ABC compare in length with the corresponding sides of DEF? of GHJ? How many triangles of the size of ABC are there in DEF? in GHJ? ABD E How A H 2. The sides of the first two triangles are in the ratio of 1 to 2, and their areas are in the ratio of 1 to 4 (the squares of 1 and 2). The sides of the first and third triangles are in the ratio of 1 to 3, and their areas in the ratio of 1 to 9. 3. Show in the same way that the sides of these squares are proportional to 1, 2, and 3, and that their areas are proportional to 1, 4, and 9, the squares of the sides. 731. 1. The corresponding sides or like dimensions of similar plane figures are proportional. 2. The areas of similar plane figures are proportional to the squares of their corresponding lines. 732. 1. If a rectangle is 4 inches long and 3 inches wide, find the width of a similar rectangle that is 8 inches long. 2. The area of a circle is 5 square inches. Find the area of a circle whose diameter is twice the diameter of the first. 3. The side of a square is 5 inches. Find the side of another square that contains 4 times as much area. 4. The sides of two regular octagons are as 1 to 3. What is the ratio of their areas? 5. A schoolroom has two square blackboards whose sides are 3 feet and 6 feet, respectively. What is the area of the first? Find the area of the second by applying the principle of similar figures. 6. A lady has two circular flower beds, one having a radius of 4 feet and the other a radius of 16 feet. How do they compare in area? 7. The sides of a triangle are 1 centimeter, 2 centimeters, and 2 centimeters. What are the sides of a similar triangle containing 25 times the area of the first? 8. If the ratio of two similar triangles is 16, what is the ratio of their bases? 9. By the principle of similar figures, find the height of a poplar tree that casts a 25-foot shadow when a boy 5 feet tall casts a shadow 6 feet long. 10. When a telephone pole 30 feet high casts a shadow of 60 feet, what is the height of a church -64 ft steeple that casts a shadow 300 feet long? 5ft. 25 ft.--- 11. Suppose that A and B are two points on the opposite sides of the pond M, and we wish to find the distance between them. Measure the distances AC and BC. Set a stake at E, a short distance from C in the line AC, and set another stake at D in the line BC, making the ratio of CD to CB equal to the ratio of CE to CA. Measure DE. The triangles DCE and ACB are then similar and AB: DE AC: CE. A B M D E 12. I wish to ascertain the distance between A and B on the opposite sides of a lake. From C, I measure the line AC, 2000 feet, and BC, 1500 feet. I set a stake at E in line with AC, 100 feet from C, and another at D in line with BC, 75 feet from C. The distance between D and E is 50 feet. distance between A and B? Similar Solids What is the 733. Solids that have exactly the same shape though they differ in volume are similar solids. The corresponding edges or other lines of similar solids are proportional. 734. 1. Draw three cubes as in these figures, having edges proportional to 1, 2, and 3. 2. How many cubes the size of the first does the second contain? the third? 3. The volumes of cubes with edges proportional to 1, 2, and 3 are proportional to 1, 8, and 27, the cubes of the edges. 735. The volumes of similar solids are proportional to the cubes of their corresponding lines. 736. 1. In what ratio are the corresponding dimensions of the two similar prisms shown here? Find the volume of the first prism (§ 721). By what number must you multiply this to get the volume of the second prism? What is the the volume of the second prism? Find the volume of the second prism by § 721 and compare results. 2. In what ratio are the corresponding dimensions of these two cones? How can you find the volume of the second cone from that of the first? Find the volume of each cone. 36 3. If a prism 5 inches high contains 30 cubic inches, how many cubic inches will a similar prism 10 inches high contain? 4. The volume of a sphere is 96 cubic feet. volume of a sphere having a diameter half as long? What is the 5. If the altitude of a cone that weighs 10 pounds is 2 feet, what is the altitude of a similar cone of the same material that weighs 270 pounds? 6. If a bowl 1.2 decimeters in diameter holds a certain quantity of milk, how many times this quantity will a similar bowl 1.8 decimeters in diameter hold? 7. A ball weighs 10 pounds. Find the weight of a ball of the same material, if its diameter is 3 times as great. 8. If a column 2 decimeters in diameter contains 728 cubic decimeters, what will be the volume of a similar column 1 decimeter in diameter? 9. How many more gallons of water can be contained in a tank 21 feet in diameter and 60 feet high than in a similar tank 40 feet high? GENERAL REVIEW 737. 1. Mr. Kirk purchased 60 shares of a Philadelphia stock company, engaged in the manufacture of infants' hose, at $63 above par, brokerage %. How much did the stock cost him? 2. The company was capitalized at $75,000. of the stock did Mr. Kirk own? What per cent 3. Mr. Kirk inspected the mill in detail. An order was received for 80 doz. pairs of infants' black hose size 4; 132 doz. size 41; 248 doz. size 5; 280 doz. size 51; 236 doz. size 6; and 112 doz. size 61. How many dozen pairs were ordered? 4. He found the weights per dozen pairs to be as follows: Size 4 41 5 14 159 Weight (ounces) Find the total weight of the yarn used for these stockings, allowing a waste in manufacture of oz. per dozen pairs. 5. The price of the yarn for this order (8321 lb.) was $1.211 per pound and for dyeing it 61 per pound. If these two items were 62% of the total cost of the stockings, what was the cost? 6. The purchaser was given his choice of buying the 1088 dozen pairs at $1.90 per dozen, or buying size 4 at $1.60 per dozen, 4 at $1.70, 5 at $1.80, 51 at $1.90, 6 at $2, and 63 at $2.10. Which was the better offer and how much? 7. He accepted the second offer and paid $ 2038 less 5% discount. How much did the mill gain, the cost being $1704.96? 8. The hose for this order were the product of 32 days' work. What was the daily output of the mill? |