180. To find the distance between two visible objects. Let P and Q be the two objects; A and B two points of observation. Measure AB. At A, measure the angles PAB, QAB. At B, measure the angles PBA, QBA. Also if the triangles PAB, QAB are not in the same plane, measure the angle PAQ. Then, in triangle PAB; AB, PAB, and PBA are known; .. AP can be calculated. In triangle QAB; AB, LQBA, and QAB are known; .. AQ can be calculated. Lastly, in triangle PAQ; AP, AQ and ▲ PAQ are known; .. PQ can be calculated. 181. In working problems, which involve points not all in one plane, it is often useful to employ a figure in which, for points not in the observer's horizontal plane, are substituted the feet of the perpendiculars from them on that plane. Thus, if P is any point not in the horizontal plane containing two points of observation A and B, draw PN perpendicular upon that plane. Then (by Art. 177) we have AN PN × (cot of elevation of P at A), Example. At A and B, the angles of elevation of an object P are observed to be a and ẞ respectively. The distance between A and B is c: and the angle between AB and the line joining A to the foot of P is 6. Find the height of P. Let N be the foot of P, and PN=x. Then in the triangle ABN we have Thus AB=c, AN=x cot a, BN=x cot ß; BAN=0. BN2 AB2+AN2 – 2AB. AN. cos 0, i.e. x2 cot2 8=c2+x2 cot2 a-2cx cot a cos 0, .. x2 (cot2 ß-cot2 a)+2cx cot a cos 0=c2, a quadratic equation giving x. EXAMPLES VIII. 1. At a point 866 ft. from the base of a column, the angle of elevation of its summit is observed to be 30°. Find the height of the column. 2. From the top of a hill the angle of depression of an object on the ground is observed to be 60°. The hill being 1732 ft. high, find the distance of the object from the point where the vertical line through the top of the hill would cut the ground. 3. A man 5 ft. 10 in. high observes that his shadow from a lamp 3 yds. 4 in. high is 1 yd. 9 in. What is his distance from the lamp? 4. A tower 50 ft. high stands on a mound; from a point on the ground the angles of elevation of the top and bottom of the tower are observed to be 75° and 45° respectively; find the height of the mound. 5. The angle of elevation of a balloon from a station due north of it is 45°, and from a station at a mile due east of the former station it is 30°. Find the height of the balloon and its distance in a straight line from the second station. 6. A man, walking along a straight road at the rate of 31 miles an hour, observes that a house, whose direction hr. ago made an angle of 45° with the road, is now directed at an angle of 75°. Find how long he will take to reach the nearest point on the road to the house and how distant the house will then be. 7. At two points, 400 ft. apart, on the bank of a straight river, the direction of an object on the other bank is observed to make with the first bank angles equal to 72° and 36° respectively. Find the breadth of the river, and the distance of the object from the first point of observation. 8. At two points A and B, at a distance c apart, the angles of elevation of an object P are observed to be a and ẞ respectively. The straight line AB is horizontal and the plane ABP is vertical. Show that the height of P above the horizontal plane through AB is с cot a ~ cot B 9. At two points A and B, at a distance c apart, the angles of elevation of an object P are observed to be a and ẞ respectively. The plane ABP not being vertical, it is observed that the difference of the angles BAP and ABP is 90°. Show that the height of P is 10. An object 26 feet high, placed on the top of a tower, subtends an angle a at a place whose horizontal distance from the foot of the tower is b feet; show that the height of the tower is b{√(2 cot a) 1}. 11. The angles of elevation of a tower from three points A, B, C in a straight line are observed to be a, ß, y respectively. If BC = a, AC = b, AB = c, show that the height of the tower is 12. A house of three equal storeys is observed from three points A, B, C in a straight line. It is found that (when looking at a fixed vertical line on the house) the angle of elevation of the top of the lowest storey at A is equal to that of the top of the middle storey at B and to that of the top of the highest storey at C. If AB=α, BC = b, show that the distance of A from the foot of the vertical line is 13. In the last question find the cosines of the angles subtended by AB, BC, and AC respectively at the foot of the vertical line; and show that if AB=2. BC and if a, ẞ are the angles subtended by AB, BC respectively, then 14. The angle of elevation of the top of a mountain from a point at its base is observed to be γ. A path leads directly from that point to the top, being for some distance inclined to the horizon at an angle a and for the remainder of the distance at a greater angle ß. At the point where the path becomes steeper, the vertical altitude is found by the barometer to be a. Show that the height of the mountain is 15. A vertical stick upon a horizontal plane. casts a shadow of length b from a lamp The horizontal and vertical distances of the bottom of the stick from its shadow are a and c respectively. If the stick subtends equal angles at the two ends of its shadow, show that the height of the lamp is abc 16. An object is observed at three points A, B, C lying in a horizontal line which passes directly underneath the object. The angles of elevation at A, B, C are a, 2a, 3a respectively. If AB = a, BC = b, show that the height of the object is 17. From a point on a hill-side of constant inclination, the altitude of the highest point of an obelisk at the top of the hill is observed to be a: a feet nearer the top of the hill it is B. Show that if be the inclination of the hill, the height of the obelisk is 18. A and B are 50 yards apart, C midway between them. From a point D, it is observed that AC subtends an angle of 60° and CB an angle of 30°. Find the distance BD. 19. A vertical tower c ft. in height stands on the top of a hill. A and B are two points in the same horizontal plane at the foot of the hill. From A the tower lies due north and the angular elevation of its base and top are a and B. From B the tower lies due west and the angular elevation of its top is 45°. Show that 20. A is the top of a tower, B and C are points in the horizontal plane on which it stands. The elevations of A at B and Care ẞ and y respectively; and the angle subtended at A by BC is a. Show that 21. A person walking along a straight road observes that the greatest angle which a building subtends is a. From this point he walks a distance c, and the front of the building is now just along his line of sight making an angle ẞ with the road. Show that the length of the building is c sin a sin ẞ sec (ẞ + a) sec 1⁄2 (ẞ − a). - 22. A man, standing at a point close to the side of a base of a pyramid whose base is square sees the sun disappear over an edge of the pyramid, half-way along it. Show that, if a and b are the distances of the man from the two ends of the side at |