s- a = 32, log cos A = 9.99552-10 log cos B = 9.98453 – 10 1. Given a 7, b=8, c=9; find A= 48° 11', = 2. Given a = 3, b=8, c = 9; find B = 61° 13', C = 73° 24'. C= 99° 36'. 19, b = 34, c=49; find A, B, C, and check by the 5. Given a = 16, c = 20; 18 b=18, c = 30; 9. Two sides of a triangle are 51 and 65. What must their included angle be in order that the third side shall be 20? AREA OF AN OBLIQUE TRIANGLE 115. The area of a triangle is found by Art. 102. 2 log K = (logs) + log (s − a) + log (s − b) + log (s — c). 1. Given a = 10, b = 16, c = 12; find K = 59.92. 2. Given a 7.9, b = 8.96, c = 10.46; find K = 34.4. = 3. Given a = 46.7, b = 84.5, c=75.6; find K = 1755.1. 4. Given a = 38.09, c = 11.2, B = 67° 55'; find K. 5. Given b= 12.5, c = 25, A = 38°; find K = 96.2. 6. Given a = 32.5, b = 56.3, C= 47° 5.5'; find K = 670. 7. Two sides of a triangle are 34.56 ch. and 48.5 ch., and the included angle is 38° 45′ 40′′. Find the area. 8. Two angles of a triangle are 38° 18' and included side is 39.5 ch. Find the area. Ans. 52.47 A. 91° 28', and the Ans. 62.88 A. 9. Two sides of a parallelogram are 59.8 ch. and 37.05 ch., and the included angle is 72° 10'. Find the area. = 10. In the quadrilateral ABCD, AB = 30 ch., BC= 25 ch., CD 37 ch., DA = 13 ch., and the diagonal AC area. = 40 ch. Find the Ans. 61.45 A. 11. Show that the area of a quadrilateral is equal to one-half the product of its diagonals into the sine of their included angle. 12. Show that the area of a regular polygon of n sides inscribed in a circle whose radius is R, is one-half the product of n by the square of R into the sine of 1 (360°). MISCELLANEOUS EXERCISES 117. Exercise XX. 1. The three sides of a triangle are 8 ch., 9 ch., and 11 ch. Find the area. Ans. 3.55 acres. 2. Two sides of a triangle are 42 ch. and 51 ch., and their included angle is 33° 18'. Find the area. 3. The diagonals of a quadrilateral are 15 and 9.8, and cross each other at an angle of 138°. Find the area. Ans. 49.18. 4. In a field ABCD, the sides AB, BG, CD, DA are 31, 47.2, 50.4, 21 rd., respectively, and the diagonal AC is 62.2 rd. Find the Ans. 1192 sq. rd. area. 5. The area of a triangle is 54.24, and two of its sides are 10.6 and 13.8. Find the angle between these sides. Ans. 47° 52'. 6. The diagonals of a parallelogram are 81 and 106, and cross each other at an angle of 29° 18'. Find the sides, angles, and area of the parallelogram. 7. Two sides of a triangle are 112.46 and 99.54, and the difference of their opposite angles is 15° 48' 32". Find the third side. Ans. 86.02. 8. The parallel sides of a trapezoid are 57 and 109, and the angles at the extremities of the latter are 53° 49' and 67° 55'. Find the non-parallel sides. Ans. 56.65; 49.35. 9. From the base of a wall the angle of elevation of the top of a pole is 23° 19', and from a window 31.5 ft. above, it is 14° 13'. Find the height of the pole. Ans. 76.42 ft. 10. From a ship the horizontal angle subtended by a round fort is 8° 20', but if the ship were to go 680 ft. nearer to the fort, it would be 10° 12'. Find the distance of the ship from the fort. 11. From a point at the foot of a mountain, the elevation of the top is 60°. After going 1760 yd. towards the top up a plane making an angle of 30° with the horizon, the angle of elevation of the top is 75°. Find the height of the mountain. Ans. 4164.2 yd. 12. A person goes 84 yd. up a slope of 2 feet in 7 from the edge of a river, and observes that the angle of depression of the edge of the water on the opposite side is 210. Find the width of the river. Ans. 530.53 yd. Find the two angles divided by drawing a 13. The sides of a triangle are 17, 21, 28. into which the largest angle of the triangle is line from its vertex to the middle point of the opposite side. 14. To find the distance between two inaccessible points, A and B, a line CD, crossing AB, is measured, its length being 600 ft. The angle DCB = 58° 12', CDB = 49° 38', ACD = 74° 16', ADC = 62° 13'. Find AB. Ans. 1151 ft. 15. One side and the opposite angle of a triangle are 309 and 21° 14.4'; if one of the remaining sides is 360, what is the other? Ans. 615.7 or 55.41. 16. Wishing to find the "high water width" (AB) of the Mississippi River directly in front of the Louisiana State University, that is, the distance from the railroad (A) on one side to the levee (B) on the other, the following measurements were made: a line AC 1000 ft. along the railroad track, BAC = 94°, 70° 41.6'. Required the width AB. = < BCA = Ans. 3575 ft. 17. The "low water width" (BC) of the Mississippi River at the same point as in Ex. 16, is 2950 ft., and from a point (4) of the university campus overlooking the river, the angles of depression of the nearer (B) and further (C) edges of the water are 7° 49.8' and 56.4'. Required the height of the university campus above the "low water." Ans. 54.96 ft. 18. In measuring a line from A to B, whose direction was N. 40° E., it was found to cross a marsh, say M. To pass around M, a distance CD=144.31 ch. was measured, starting from a point C of AB, and making with AB an angle BCD=19° 54.4'. At Dit was found that a line DE, making an angle CDE = 140° 10.3' with CD, would cross the line AB at some point, say G, beyond M. Find the distances DG and CG, and also the angle DGB in order that the course of AB may be resumed. Ans. DG = = 143.98 ch.; CG = 271.06 ch.; DGB=160° 3.7'. 19. When the elevation of the sun is 48°, a pole, standing vertically on a slope whose angle with the horizon is 15°, casts a shadow directly down the slope 44.3 ft. long. How high is the pole? Ans. 30.06 ft. 20. Three circles whose radii are 12, 16, and 25 are tangent exter = 21. The three points A, B, C are in a horizontal plane, and CD is a vertical pole; AB=435.53 ft., CAB = 140° 40.2', CBA 10° 7.6', CAD 32° 45.6', CBD = 10° 7.3'. Find the height (CD) of the pole. Ans. 101 ft. = 22. Between the points A and B, a distance of 13 miles, the grade of a road on a straight line would be 1 foot in 40. What would be the length of a zigzag road from A to B that would have a grade of 1 foot in 70? 23. An engineer, desiring to know the distance between two hostile forts (A, B) on the same horizontal plane, goes up in a balloon to the height of 517.3 rd., and from that point (C) observes that the angles of depression of A and B are 21° 9' 18'' and 23° 15′ 34'', and that the angle ACB = 15° 13' 15". Find the distance between the forts. Ans. 383.35 rd. = 24. The four sides of a city block are AB: 423.24, BC= 162.36, CD = 420.80, and DA 160.60 ft., the first two sides being perpendicular to each other. Find the angles between the other sides. = 25. On the bank of a river there is a column 200 ft. high supporting a statue 30 ft. high; the statue to an observer on the opposite bank subtends an equal angle with a man 6 ft. high standing at the base of the column. Required the width of the river. Ans. 10√115 ft. NOTE. After the student has learned to compute readily by natural functions and logarithms, it will be sufficient to give only the statements in solving problems like the two following. 26. Wishing to determine the distance between two objects (A, B) on the opposite side of a river, I lay off a line, CD 800 ft., along the river, C being nearly opposite A; and then find by measurement that ACB = 57° 20', ACD = 94° 20', ADB = 53° 20', BDC = 98° 35'. Find the required distance AB. = 27. Given AB = 800 yd., AC = 600 yd., BC 400 yd., ADC = 33° 45', BDC= 22° 30'. Required AD, CD, BD. AD=710.15 yd., CD=1042.5 yd., BD=934.28 yd. HINT. — Describe the circumference through A, B, D, and draw AE, BE; then EAB = BDC, and EBA = ADC. Find ABC, Art. 113, then / EBC, then EB, then ▲ C, Art. 111, etc.. |