24. Draw a triangle and measure the lengths of its sides to hundredths of an inch by use of the diagonal scale. In measuring, apply the compasses to a side of the triangle, then to the diagonal scale. 25. Draw a square with a side 1 in. long. Draw a diagonal. By use of compasses and diagonal scale, measure the length of the diagonal to hundredths of an inch. Find the ratio of the diagonal to a side of the square. 26. Draw a line-segment 1.23 in. long. One 1.07 in. long. One 0.89 in. long. 27. Construct an equilateral triangle whose side is 1.37 in. 28. Construct a triangle whose sides are 1.62 in., 1.74 in., and 1.95 in. 29. The square OPMN is said to be inscribed in ∆ABC. By studying the figure, explain how to construct the square OPMN in the given triangle, and give the proof. C F N M SUGGESTION. — DEFC is a square, and AF meets BC at M, determining one vertex of the A OD PE B 131. The practical measurement of distances. - The proportion between the sides of similar triangles affords a useful means of finding distances in practical work where direct measurements are impossible. Some applications are shown in the following problems. EXERCISES 1. The fact that the heights of two objects are to each other as the 2. A stick EF held vertically casts a shadow DE which is 34 ft. long. At the same time a tree BC casts a shadow AB which is 50 ft. long. If EF = 6 ft., find BC. A B 3. The distance AB across a stream is obtained as follows: A line AC is run off at right angles to AB, along the line CD is measured off at The point E of AC which AE E shore. From C the right angles to AC. is in line with D and B is then located. and EC are measured. AB is then computed by D C proportion. Write the proportion between AB and the measured distances. Prove this proportion. 4. In the figure of Ex. 3, if AE = 120 ft., EC = 30 ft., and CD = 40 ft., find AB. 5. The following method may be used for estimating the distance from the observer to an inaccessible object: A With the left eye closed, the finger is pointed, at arm's length (at 0), toward the object A. Then without moving the finger, the right eye is closed and left eye opened, D C B when the object appears to have moved to B. The distance AB through which it appears to have moved, being transverse to the line of sight, is estimated. The distance from the finger O to the object is approximately 10 times the distance AB. The distance CO from the eye to the outstretched finger of the average person is approximately 10 times the distance CD between the eyes. Prove, then, that OA = 10 AB. If the object appears to have moved 1100 ft., how far away is it? How many miles? 6. The U. S. S. Texas (1915) is 565 ft. long. When observed at sea by the method of Ex. 5, it appeared to move through a distance of four ship lengths. How far away was it? How many miles? C E D 7. A method employed several centuries ago, before modern instruments were invented, for determining the distance from A to an inaccessible point B was as follows: Upon a vertical staff AC was placed an instrument resembling a carpenter's square. The blade CD was pointed toward B, and at the same time the point Fon the ground at which the blade СЕ F A B pointed was marked. FA and AC were measured. Then AB was com 8. In Ex. 7, if FA = 2 in., and AC = 62 in., find AВ. NOTE. - The student would find it interesting and profitable to use the different methods suggested in the preceding exercises in determining the heights and distances of objects or points in the neighborhood of the school. 9. The distance between two accessible points A and B which are separated by an obstacle may be measured as follows: From a convenient point P the distances PA and PB are measured. Then in these lines points Cand D, respectively, are located such Prove the proportion by means of p which AB may now be computed. C If PA = 840 ft., PC = 120 ft., CD = 400 ft., find AB. A 10. Before modern instruments were invented, an inaccessible dis toward A. Show how it was possible from these measurements to compute AB. If BC = 200 yd., bc = 10 in., and ba = 16 in., compute AВ. 11. The cross-staff (See Ex. 6 and Note, page 53) may be used to find the height AB of an object as follows: The horizontal cross-bar DE is raised or lowered on the staff FG until D, F, and B fall in a straight line. Then DE, EF, GE, and Explain how AB may be computed. If DE = 18 in., EF = 24 in., GE = 36 in., and D GA = 60 ft., find AB. B F C 12. By the aid of the diagram explain how the cross-staff may be used to obtain the horizontal distance from a point A to an inaccessible point B. C D E If AC = 60 in., EC = 14 in., and ED A = 18 in., find AB. B 13. The geometric square is another instrument used in practical meas urements before modern engineering instruments were invented. It consists of a square frame, along two ad jacent edges of which is marked a scale, and from the opposite corner of which a plumb line is suspended. A pair of sights on another edge aid in pointing the instrument. When the height MN of an object is to be found, the square is held in a vertical plane and the edge bearing the sights is pointed toward N. The point C where crosses the scale is the plumb line then noted. The height of the instrument DA and the horizontal distance DM are measured. PN is computed by proportion. 14. In the figure of Ex. 13, if AB = 48, CB = 20, DM = 96 ft., and DA = 5 ft., find MN. 15. The horizontal distance PQ may be measured by means of the geometric square as follows: The square is held directly over P and the edge bearing the sights directed toward Q. The point C where the plumb line then crosses the scale is noted. The height of the instrument AP is measured. Then PQ is computed by proportion. Prove the proportion by means of which PQ may be computed. A B P If AP = 4 ft., AB = 48, and BC = 11, find PQ. Q NOTE. - It will be found easy and interesting for the student to make a geometric square and use it in such measurements as those described in the exercises above. 16. A pantograph is an instrument for drawing a figure similar to a given figure, and is useful for enlarging or reducing maps and drawings. It consists of four bars, parallel in pairs and jointed at B, C, D, and E. A turns on a fixed pivot, and pencils are carried at D and F. BD and DE are so adjusted as to form a parallelogram BCED and such that any AB required ratio is equal to AC CE CF C E B A D F Show that A, D, and Fare always in a straight line. AD 17. Show that in the pantograph of Ex. 16 the ratio remains AF AB constant and equal to so that if the pencil Ftraces a given figure, AC' the pencil D will trace a similar figure. 132. Trigonometric ratios. - Let ∠ BAC be any acute angle. That is, the ratios of the (1) altitude to the hypotenuse, (2) base to the hypotenuse, and (3) altitude to the base are the same regardless of the length marked off along the side AC. This is expressed by saying that for any given ∠BAC, the three ratios are constant. These three constant ratios which are connected with any given acute angle are called trigonometric ratios. |