angle. 6. Given k, l, the diagonals of a parallelogram, and α, the included Show that the area of the parallelogram iskl sin a. 7. If the diagonals of a parallelogram are 24 ft. and 32 ft., and an included angle is 45°, find the sides and the area. 8. Two observers 5.0 miles apart on a plain, and facing each other, find that the angles of elevation of a balloon in the same vertical plane with themselves are 58° and 65°, respectively. Find the distances from the balloon to the observers, and also the height of the balloon above the plain. Ans. Distances from observers, 5.1 m. and 5.4 m.; height above plain, 4.6 m. a 65° B C 58 D A C=5M 9. A lighthouse was observed from a ship to bear N. 35° E.; after the ship sailed due south 4 miles, it bore N. 18° E. Find the distance from the lighthouse to the ship in both positions. NOTE. The phrase "to bear N. 35° E." means that the line of sight from the ship to the lighthouse is to the east of a line running due north from the ship, and makes with the latter line an angle of 35°. 175' 15' 225 C B 10. ABCD is a plat of ground in the shape of a trapezoid with AB and DC the bases. Find the area, if AB 225 ft., AD = 100 ft., DC= 175 ft., and LBAD 78° 15'. = 11. A railroad having a hundred feet right of way cuts through a farmer's field as shown in the figure. If the field is rectangular, and the measurements made are as shown, find the number of square rods occupied by the right of way; also find the assessed damages if the land is appraised at $100 an acre. 12. In a preliminary railroad survey the distance between two points A and B on opposite sides of an 29 30 400 impenetrable swamp and the line of direcBtion of AB were computed from the following data: Two points C and D visible E from each other, and visible from A and B, respectively, were chosen, and distances and angles measured as shown. Find the distance AB; also CAB and ABD. F G SUGGESTION: 1. Draw DF and BG LAC, produced, and draw DE || AC. REGULAR POLYGONS AND CIRCLES 472. PROBLEM. Given R, the radius of a circle, to find the side SOLUTION: 1. Let AB a be a side of the inscribed equilateral triangle. Join the center O to B, and draw OM LAB, cutting 1. If the radius of a circle is 1, find the side, the apothem, and the area of an inscribed equilateral triangle. 2. If the side of an equilateral triangle is a, show that the radius of the 3. If the radius of a circle is R, show that the side of an inscribed square is R√2. = SUGGESTION: Let AB a be a side of the inscribed square. Why? 4. If the radius of a circle is R, show that the area of the inscribed square is 2 R2, and the apothem is R√2. 2 475. PROBLEM. Given R, the radius of a circle, to find the side of an inscribed regular dodecagon. E M B FIG. 215. SOLUTION: 1. Let AB be a side of an inscribed regular hexagon. Draw the diameter CE, bisecting arc AB, and cutting AB in M. REMARK. By an exactly similar method, when the side of any inscribed regular polygon is known, the side of an inscribed regular polygon of double the.number of sides can be found. 1. If the radius of a circle is 1, find the side, the perimeter, and the area of an inscribed regular dodecagon. 2. Find the side of a regular octagon inscribed in a circle, if (1) the radius is 1; (2) if the radius is R. 477. PROBLEM. Given R, the radius of a circle, to find the SOLUTION: 1. Let AB be a side of an inscribed regular hexagon. Draw OM'LAB, and draw A'B' tangent to the circle at M', intersecting OA and OB, produced, at A' and B', respectively. 2. Then A'B' is a side of a circumscribed regular hexagon. Why? REMARK. By an exactly similar process, when the side of any inscribed regular polygon is known, the side of a circumscribed regular polygon of the same number of sides can be found. 1. If the radius of a circle is 1, find the perimeter and the area of a circumscribed regular hexagon. 2. If the radius of a circle is R, find the side of a circumscribed equilateral triangle. 3. If the radius of a circle is 1, find the perimeter and the area of a circumscribed regular dodecagon. Use the results of Ex. 1, § 476. 4. If the radius of a circle circumscribed regular octagon. is 1, find the perimeter and the area of a Use the result of Ex. 2, § 476. 479. Regular polygons in general. We have seen that as soon as the side of any inscribed regular polygon is known, the sides of inscribed and circumscribed regular polygons of double the number of sides may be found. Thus, beginning with the square, the sides of regular polygons of 4, 8, 16, 32, ..., 2 sides may be found; beginning with the equilateral triangle, the sides of regular polygons of 3, 6, 12, 24, ..., 3 x 2" sides may be found. MEASUREMENT OF THE CIRCLE regular polygons F F E A B 480. Consider ABCDEF and A'B'C'D'E'F", respectively, inscribed and circumscribed about a circle. Imagine the number of sides. indefinitely increased, by any law - as by successive doubling. Then it is clear that the successive perimeters of the inscribed polygons become greater and greater (why?), and the successive perimeters of the circumscribed polygons become smaller and smaller (why?). Hence the difference between the perimeters of the inscribed and circumscribed polygons becomes less and less, though never zero (why ?). B' We shall assume 481. FUNDAMENTAL PROPOSITION. that the perimeters of regular polygons inscribed in and circumscribed about a circle approach a common limiting value, when the number of sides is indefinitely increased; and also that the areas of these polygons approach a common limiting value. 482. The circumference and the area of a circle. The common limit of the perimeters of regular polygons inscribed in, and circumscribed about, a circle, when the number of sides is indefinitely increased is called the length of the circle, or its circumference; and the common limit of their areas is called the area of the circle. |