1 chain 100 links = 4 poles = 22 yards. 1 degree 69 English miles = 60 geographical miles. PROBLEM 1. Any two sides of a right angled triangle being given, to find its third side. 1st. The lines A B, B C being given, to find A C. Rule.-Square each of the sides, and add them together; the square root of this sum will be the side A C. 2nd. The lines A C, A B being given, to find B C. Rule.-Square the lines A C and A B, and take their difference; the square root of this difference will be the side B C. 3rd. The lines A C, B C being given, to find A B. Rule.-Square the lines A C and B C, and take their difference; the square root of this difference will be the side A B. Therefore B C = √ A C2 — A B2 = √542 EXAMPLES. Section 1. 39237 ft. 4 in. 1. Given B C = 50 feet, A B = 80 feet; find A C. 8. Given AC =56 feet, A B = 16% feet; find B C. 9. Given AB = 585 feet, B C = 857 feet; find A C. Section 4. 10. Given B C = 535 feet, A C = 553,5 feet; find A B. 11. Given B C 4 feet, A B = 3 feet; find A C. 12. Given B C = 2 A B ; required A C, when A B = 10. Section 5. 13. The sum of A B and B C is 36 feet, and their difference is 14 feet; required A C. 14. A sheer leg, whose length is 104 feet, is inclined to the horizon, so that the distance from its base to the foot of a perpendicular from its top is 26 feet required its height. Section 6. 15. A ladder, 54 feet long, is placed with one end against an upright wall, and the other at a certain distance from the foot of the wall; the ladder is then observed to slide 15 feet on the ground when its top reaches the ground; required the height of the wall. 16. The side of a square is 84 feet; required the length of its diagonal. Section 7. 17. The length of a beam of a ship is 24 feet 6 inches, and the distance of the perpendicular from the middle of the beam to the apron is 14 feet 8 inches; required the length of the horning lines. 18. The perpendicular on the stem of a vessel is 30 feet 3 inches, and a distance of 25 feet 6 inches is set up on the perpendicular; find the hypothenuse. The following require the application of Algebra. Section 8. 19. A ladder, 15 feet long, is placed with one end against an upright wall, and the other on a horizontal line; while the upper end falls to a height of 3 feet the lower one slides 3 feet; find the height of the ladder in its first position. 20. A ladder, 45 feet long, is placed with one end against an upright wall, and the other on a horizontal line; the foot of the ladder is observed to slide so that its distance from the wall is now doubled, and the height of the top from the foot of the wall is now the same distance as the foot of the ladder was from the wall in its first position; how far has the top slid? 21. A ladder, 20 feet long, is placed with one end against an upright wall, and the other on a horizontal line; required the position of the ladder so that when the foot of it slides 2 feet on the ground the top may slide 1 foot down the wall PROBLEM 2. Given, the three sides of a triangle; to find the perpendicular from any angle to its opposite side. Rule.-Add the sides A C and B C together, and call this the first sum. Subtract the line B C from A C, and call this the difference. Multiply the sum and dif ference together, and add the product to the square of the side A B. Divide this last sum by twice the side A B, and the quotient will be the segment A D. Then, the triangle A CD being a right angled triangle, the perpendicular C D can be found by Problem 1. The three perpendiculars A F, B E, C D will always pass through the same point. BD2 And, C D = √62 AD = √a2 The segments A D, B D, may also be found by the following rule, AB: AC+ B C :: A C-BC: AD — B D. One half of this difference being added to, and subtracted from half the line A B, will give the segments A D and B D. 1. The base A B = EXAMPLES. Section 1. 16 feet, A C = 11 feet, and B C 8 feet; required the segments and perpendicular on A B. 2. The two sides of a triangle are 2 and 3, and the base 4; reguired the segments and perpendicular with respect to each angle. Section 2. 3. The three sides of a triangle are 14, 9, and 12 feet; required the segments and perpendicular with respect to each angle. Section 3. 4. The three sides of a triangle are 24, 18, and 15 feet; required the segments and perpendicular with respect to each angle. Section 4. 5. The three sides of a triangle are 8, 7, and 6 feet; required the segments and perpendicular with respect to each angle. PROBLEM 3. In two similar figures there is given two corresponding dimensions, one in each figure, and also another dimension in one figure; to find its corresponding dimension in the other. N.B.-"Similar figures are those which have their several angles equal, each to each, and the sides about the equal angles proportional." (Euclid, vi.) The same proportion holds in respect to any other corresponding lines in the similar figures. Example 18.-Section 6. 28 miles per hour 28 x 5280 147840 feet per hour. Then, 1071 7: 147840: height required 966 27 feet. |