and the length of DE (6 units) were known. Then, from the following proportion, the length of FE could be found: 1. Two sides of a triangle, AB and BC, are 5 ft. and 8 ft. respectively. A similar triangle has a side AB, 15 ft. long. How long is its side BC? 2. One side of a triangle is 2 in.; the corresponding side of a similar triangle is 5 in. Another side of the first triangle is 7 in. What is the corresponding side of the second triangle? 3. Some boy scouts staked off a line 200 ft. long on the bank of the Hudson River. They measured the angles formed by sight ing a station on the opposite shore from each end of the 200-ft. line. Then they drew a similar triangle on paper, using 1 in. for each 200 ft. They found the altitude of the small triangle to be 22 in. What was the altitude (distance across the river) of the large triangle? (The altitudes of two similar triangles have the same ratio as any two corresponding sides.) 4. To find the distance from A to B through the building shown in the picture below on this page, drive a stake at C, where both A and B are visible. The point C is found to be 24 yd. from A and 15 yd. from B. Extend BC 5 yd. and AC 8 yd. beyond C, so that the line EF will be parallel to AB. When EF is measured, it is found to be 10 yd. The triangles ABC and CEF are similar and have proportional sides. = 6. AC 15 yd.; CF = 3 yd.; EC = 5 yd.; CB = 25 yd.; EF = 5 yd. What does AB equal? *7. The boy scout in the picture is measuring the height of a tree by the use of a yard stick and a cardboard that has a peep-hole in it. He stands 25 ft. from the tree, holds the yard stick exactly 1 ft. from his eye, and sights the top and the foot of the tree through the peep-hole in the cardboard, which is held by strings tied around his head. He observes that a line from the top of the tree passes through the 3-in. mark on the yard stick, and that a line from the foot of the tree passes through the 18-in. mark. Thus a triangle is formed whose altitude is 1 ft. and whose base is ·24 Measuring the Height of a Tree 15 in. (18 in. - 3 in.), or 14 ft. At the same time, a large triangle is formed whose altitude is 25 ft. and whose base is the height of the tree. These two triangles are similar because their corresponding angles are equal, and their corresponding sides proportional. *8. If the lines of sight from the top and the foot of a tree pass through the 3-in. and the 18-in. mark, when a boy is standing 40 ft. from a tree with the yard stick 1 ft. in front of him, how high is the tree? (Make drawing.) == *9. Distance from the tree 30 ft.; lines of sight pass through 4-in. and 28-in. marks on a yard stick held 1 ft. from the eye. Find the height of the tree. 10. One boy in the picture is holding a 5-ft. pole. The other boy finds its shadow to be 2 ft. long. They find that the telegraph pole beside them casts a shadow 20 ft. long. Two similar triangles are thus formed whose sides are proportional. 11. A telegraph pole casts a shadow 15 ft. long when a 5-ft. stick held in Measuring the Height of a Telegraph a similar position casts a shadow 2.5 ft. long. How high is the pole? Pole by its Shadow 12. The length of the shadow of a tower is 100 ft. at the same time that a 6-ft. stick casts a 10-ft. shadow. What is the height of the tower? *13. Bring to class a problem based upon your own measurements of shadows. 1. What do the vertical lines in this graph show? The horizontal lines? 2. About what was the population of the country for each year represented in the graph? *3. Redraw the graph so as to include the population for last year. *4. Write a brief description of a graph and read it to the class. The best description may be written on the board. 2. Different Forms of Graphs Many different forms of graphs are used to picture facts. Answer the questions about the following graphs. |