3. In a triangle, b = 100 ft., A 50°, C= 60°; 6. In a triangle the perimeter is 20 ft., and the angles are 50°, 60°, and 70°, respectively; required the area. Ans. 18.85 sq. ft. 7. In a triangle the perimeter is 60 ft., and the radius of the inscribed circle is 5 ft.; required the Ans. 150 sq. ft. area. 162. Problem. To find the area of a quadrilateral. 1. When two opposite sides and the perpendiculars to these sides from the vertices of the angles at the extremities of a diagonal are given. Let b and b' be two opposite sides, and a and a' the perpendiculars to these sides from the vertices of the angles D and B. A D C B Ն ABCD=k, ABD ab, DCB = 1⁄2 a'b'. .. (1) kab + 1⁄2 a'b'. Corollary 1.-If ' is parallel to b, the quadrilateral becomes a trapezoid, a' a, and (1) becomes (2) k a (b + b′). - = Corollary 2. If b' b, the trapezoid becomes a parallelogram, and (2) becomes Corollary 3. If b' 0, the trapezoid becomes a triangle, and (2) becomes (4) kab. 2. When a diagonal and the perpendiculars to the diagonal from the vertices of the opposite angles are given. Let d denote the diagonal, and p and p' the perpendiculars. B a A p' D ADCdp'. .. (5) k= 1 d (p + p′). 3. When the sides and a diagonal are given. Let the areas of the triangles be denoted by k' and k", which are found by article 160, (5). .. (6) kk' + k". 4. When the sides and one angle are given. Draw the diagonal opposite the given angle, and call the areas of the triangles k' and k". In one triangle we have two sides A and their included angle, from which we find the area and the diagonal. Then, in the other triangle, we have the three sides, from which we find the area. .. (7) kk' + k". 5. When the diagonals and their included angle are. given. Let d and d' denote the diagonals E p and q, r and s their segments, and A their included angle. B The angles at A are equal or supplementary; hence their sines are equal. BCDE = BAC+ CAD + DAE + EAB. 8 D BCDEk, BAC=1ps sin A, CAD=98 sin A. DAE gr sin A, EAB = pr sin A. .. k But psqs+ qr + pr (p + q) (r + 8) = dd'. 6. When the angles and two opposite sides are given. supplementary, their sines are equal. The same is true of the angles at D. 7. When three sides and their included angles are given. Let a, b, and c be the given sides, and A and B their included angles. ABCD= ABD + DBC. ABCD k, ABD = ab sin A. Find B' and d, B" BB', DBC=cd sin B". = .. (10) kab sin A+ cd sin B". 8. When the sides of a quadrilateral inscribed in a circle are given. Let a, b, c, d be the given sides. sin CV1-cos2C, Let s a + b + c + d. ... (11) k=V (38 − a) (18—b) († 8 —c) (48 — d). 163. Examples. 1. Two opposite sides of a quadrilateral are 35 rds. and 25 rds., and the perpendiculars to these sides from the extremities of the diagonal are, respectively, 12 rds. and 16 rds.; required the area. Ans. 410 sq. rds. 2. Find the area of a trapezoid whose bases are 15 rds. and 20 rds., and whose altitude is 18 rds. Ans. 315 sq. rds. 3. Two adjacent sides of a parallelogram are 30 rds. and 40 rds., and their included angle is 30°; required the area. Ans. 600 sq. rds. 4. The diagonal of a quadrilateral is 40 rds., and the two perpendiculars to the diagonal from the vertices of the opposite angles are 10 rds. and 15 rds., respectively; required the area. Ans. 500 sq. rds. 5. The sides of a quadrilateral are 30 rds., 40 rds., 50 rds., and 60 rds., and the diagonal drawn from the intersection of the sides, whose lengths are 30 rds. and 40 rds., is 70 rds.; required the area. Ans. 1874.22 sq. rds. 6. The sides of a quadrilateral are 25 rds., 35 rds., 45 rds., 55 rds., and the angle included by the sides, whose lengths are 35 rds. and 45 rds., is 50°; required the area. Ans. 927.47 sq. rds. 7. The diagonals of a quadrilateral are 30 rds. and 40 rds., and their included angle is 30°; required the area. Ans. 300 sq. rds. 8. The angles of a quadrilateral are 80°, 110°, 88°, 82°, the side included by the first and second of these angles is 25 rds., and the side included by the third and fourth angles is 45 rds.; required the area. Ans. 4105.08 sq. rds. |