6. A man starting from a point P walks 60 yd. west and then 35 yd. north. What is his direct distance from P? Suggestion: On a scale drawing to the right is east, and to the left is west. 7. In building a barn a carpenter aims to make the height of the A -------------- -36FIG. 162 roof (Fig. 162) equal to one-fourth of the width of the building. By means of a scale drawing find the angle between the rafter AC and the B plate AB. 8. Two automobiles leave a garage at the same time. One running at a rate of 20 miles an hour travels east for two hours, and then north for one hour. The other running at a rate of 25 miles an hour travels southwest for one hour and north for two hours. Find the distance between them at this time, using a scale drawing. Suggestions: Make a rough sketch before attempting an accurate drawing. The required line is to be the last line drawn. Southwest means halfway between south and west (p. 89). 9. Solve Exercise 1 (§76) by means of a scale drawing. The measured parts are given in Fig. 163. Suggestion: Draw CA first. 10. Solve Exercise 5 (876) by means of a scale drawing, the measured parts being shown in Fig. 164. Suggestion: Draw AB first. Check the accuracy of the drawing by finding the third angle. 11. A surveyor wishes to find the distance AB (Fig. 165) between two points, A and B, on opposite sides of a river. B is so located that it cannot be seen Similarly, 200 ft. from A he locates D, and finds by measurement that angle ADB=36°. Find the distance from A to B. 81. Angle of elevation. To find the height of a chimney AB (Fig. 166) a surveyor places his transit at a point C taken at a convenient distance from A. the direction EB. Angle DEB is called the angle of elevation of the point B from the point E. EXERCISES 1. The telescope (Fig. 166) is 4 ft. above the ground and 50 ft. from A. AC is in horizontal position. Measure the angle of elevation, make a scale drawing of AEDB, and from it find the length of DB. Find AB. 2. The angle of elevation of the top of a tree is 30° when observed at a point 40 ft. from the foot of the tree. How high is the tree above the horizontal line of sight? To test the accuracy of the drawing use §66, Exercise 11, i.e., measure the hypotenuse of the triangle and compare its length with the length of the side opposite to the 30° 55' FIG. 167 45' angle. 3. A flagpole (Fig. 167) 45 ft. high casts a shadow 55 ft. long. 4. When the angle of elevation of the sun is 30° a building casts a shadow 85 ft. long. Find the height of the building. Can you state a conven 75 FIG. 168 ient way of checking the accuracy of your drawing? At 5. On top of a building (Fig. 168) is a tower. a point A, 75 ft. from the base of the building, the angle of elevation of the top of the tower is 35°, and the angle of elevation of the base of the tower is 20°. Find the height of the tower. D D 6. To determine the height of a tower AB (Fig. 169) a surveyor C B places his transit at a point C and finds the angle of elevation ECB to be equal to 68°. He next places the transit at a point D in line with C and E and 60 ft. from C. He finds the angle of elevation ED1B to be 35°. If in both cases the telescope was 3 ft. above the ground, find the length of AB. 82. Angle of depression. A transit is placed on top of a cliff A (Fig. 170) overlooking a river. The telescope is first pointed horizontally in the direction AC. It is then turned through angle CAB until it points to a passing boat B. The angle CAB is called the angle of depression of the boat from the point A. Show that the angle of depression of B from the point A is the same as the angle of elevation of A from the point B. EXERCISES 1. From the top of a lighthouse 100 ft. high (Fig. 171) the angle of depression of a boat is 50°. How far is the boat from the top of the lighthouse? 2. An observation balloon B is anchored 2000 yd. above a point A. The balloonist observes the enemy at a point C and finds the angle of depression of C from B to be 62°. How far is it from A to C? 50° 50° FIG. 171 ·100' 3. From the top of a vertical cliff 100 ft. high the angle of depression of a buoy is 30°. Find its distance from the bottom of the cliff. THE SIMILAR-TRIANGLE METHOD OF FINDING UNKNOWN DISTANCES 83. Advantages of the method. The use of the congruent-triangle method of finding distances is limited because it requires a triangle to be laid off which is congruent with the triangle containing the desired distance as a side. The scale-drawing method is an improvement because the required triangle is drawn to scale on paper, not actually on the ground. However, the errors introduced in making a drawing, and the time spent in attempting to attain a high degree of accuracy in drawing and measuring, offer serious objections to this method. The "method of similar triangles," which is the next to be studied, has the advantage that an exact drawing is not needed, a rough sketch being sufficient. Furthermore, the final result is not determined by measurement, but by solving an equation, which makes greater accuracy possible. 84. Similar triangles. On squared paper draw a triangle, as ABC (Fig. 172). Draw A1 B1 not equal to AB. 1 1 On segment A,B, construct triangle A,B,C1 so that angle A = angle A1, and angle Bangle B1. Tell, without measuring, how angle C should compare with angle C1. 1 Measure angles C and C1 to test the accuracy of the drawing. How do the two triangles compare as to shape? As to size? |