66. Area of a triangle. Suppose that the area of a triangle ABC is required. Let the length of the perpendicular DC from C to AB, or AB produced, be denoted by p, and let the area be denoted by S. The following cases may occur: I. One side and the perpendicular on it from the opposite angle known, say (c, p). S=1cp. [By geometry.] (1) II. Two sides and their included angle known, say, b, c, A. (See Figs. 62 a, 62 b.) S = { cp = c · AC sin BAC. [Art. 40.] .. Sbc sin A. [Compare Art. 31.] (2) That is, the area of a triangle is equal to the square root of the product of half the sum of the sides by the three factors formed by subtracting each side in turn from this half sum. See Art. 34 a for another derivation of this formula. IV. One side and the angles known, say, a, A, B, C. EXAMPLE. Write and also derive the similar formulas in b and c. EXAMPLES. 1. Find the areas of the triangles in Exs. 1-5, Art. 61. 2. Find the areas of the triangles in Exs. 1-5, Art. 62. 3. Find the areas of the triangles in Exs. 2, 3, Art. 60. 67. Area of a quadrilateral in terms of its diagonals and their angle of intersection. Similarly, ={(AL+LC)DL sin ALD, (since sin CLD=sin ALD) =AC · DL sin ALD. area ABC = { AC · BL sin BLC = { AC · BL sin ALD. .. area ABCD = AC(DL+LB) sin DLA= AC · BD sin DLA. .. area of a quadrilateral is equal to one-half the product of the two diagonals and their angle of intersection. EXAMPLES. 1. Find the area of a quadrilateral whose diagonals are 108, 240 ft. long, and inclined to each other at an angle 67° 40'. Find the sides and angles of a parallelogram having these diagonals. 2. So also when the diagonals are 360, 570 ft. long, and their inclination is 39° 47'. 3. The diagonals of a parallelogram are 347 and 264 ft., and its area is 40,437 sq. ft. Find its sides and angles. 4. Solve an isosceles trapezoid, knowing the parallel sides a=682.7 metres, c = 1242.6 metres, and the non-parallel equal sides b = d = 986.4 metres. Find the angles, the area, the lengths and angle of inclination of the diagonals. 68. The circumscribing circle. Let the radius of the circle described about a triangle ABC be denoted by R. It has been shown (see equation (2), Art. 54) that That is, the radius of the circumscribing circle of any triangle is equal to half the quotient of any side by the sine of the opposite angle. i.e. Area BOC+ area COA M (2) Draw A N B FIG. 64. + area AOB = area ABC. ·· 1 ar + 1 br + 1⁄2 cr = √s(s − a)(s — b)(s — c), or S. That is, the length of the radius of the inscribed circle of a triangle is equal to the number of units in its area divided by half the sum of the lengths of its sides. See reference in Art. 62. NOTE. Formula (8), Art. 62, can be readily derived from Fig. 64. By geometry, AN = MA, BL = NB, CM = LC. 70. The escribed circles. L M A B N i.e. An escribed circle of a triangle is a circle that touches one of the sides of the triangle and the other two sides produced. Letra denote the radius of the Area ABQ area CAQ — area BCQ = area ABC. Draw Other interesting relations between the sides, angles, and related circles, of a triangle, are indicated in the exercises in the latter part of the book. EXAMPLES. 1. Find the radii of the circumscribed, inscribed, and escribed circles of some of the triangles in Arts. 55-58. 2. Find the radii of these related circles of some of the triangles in Exs. 1-3, Art. 66. N.B. Questions and exercises suitable for practice and review on the subject-matter of this Chapter will be found at pages 193, 194. CHAPTER IX. RADIAN MEASURE. 71. The radian defined. The system of measuring angles with a degree as the unit angle, was described in Art. 11. Since the time of the Babylonians this system has been the common practical method employed. Another method of measuring angles was introduced early in the last century. This method is used to some extent in practical work, and is universally used in the higher branches of mathematics. It is employed, on account of its great convenience, in the larger and more important part of what is now called trigonometry, namely, the part which is not concerned with the measurement of lines and angles, but which pursues investigation of the properties of the quantities that, so far in this book, have been called the trigonometric ratios. A very little knowledge of the trigonometric ratios is sufficient for the solution of triangles. The more detailed and extended study of angles and their six related numbers, constitutes part of what is sometimes called Higher Trigonometry, but, more generally, Analytical Trigonometry. This subject is a large one, and has close connections with many other branches of modern mathematics. |