PROPOSITION XXXVI. THEOREM 331. In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it. Given in ▲ abc, p the projection of b upon c, and the angle opposite a an acute angle. Proof. Denote the perpendicular upon c by h. In the figure on the left [To be completed by the student.] 332. REMARK. The equation a2b2+c2-2 cp contains four quantities. Therefore any one of them may be found by algebraical methods if the other three are given. (Similarly in the following propositions.) Ex. 958. In ▲ abc, find a if (a) b 8, c = 5, p = 4. (c) b = 5, c = 6, p = 3. (b) b = 24, c = 9, p = 12. Ex. 959. If two sides of a triangle equal 15 and 25 respectively, and the projection of 15 upon 25 equals 9, what is the value of the third side? Ex. 961. The sides of a triangle are 13, 14, and 15. tion of 13 upon 14. Ex. 962. The sides of a triangle are 5, 7, and 8. of 8 upon 5. Ex. 963. The sides of a triangle are 10, 17, 21. of 10 upon 21. [See practical problems, pp. 298 and 299.] Find the projec Find the projection Find the projection PROPOSITION XXXVII. THEOREM 333. In any obtuse triangle, the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides, increased by twice the product of one of these sides and the projection of the other side upon it. Given in ▲ abc, p the projection of b upon c, and the angle opposite a obtuse. Ex. 964. In ▲ abc, b = 6, c = 10, p = 3, and ≤ A is obtuse; find a. Ex. 965. In ▲ abc, b = 10, c = 9, p = 6, and ZA is obtuse; find a. Ex. 966. In ▲ abc, find a if (a) b=3, c = 5, ▲ A = 120°. (b) b = 8, c = 7, ZA: = 120°. (c) b= 16, c5, A = 120°. (d) b = 24, c = 11, A = 120°. 334. REMARK. If we consider the projection of one side of a triangle upon another as positive when the projection lies on that line, but as negative when it lies on the prolongation, Props. XXXVII and XXXV become special cases of Prop. XXXVI, and we have always: a = b + c − 2 ep. To compute the projections of sides of a triangle whose angles are not known, always apply this equation. If the result is negative, the triangle is obtuse. Ex. 967. of b upon c. Ex. 968. b upon c. In ▲ abc, a = 20, b = 15, and c = = 7. Find the projection In A abc, a = 20, b = 15, c = 25. Find the projection of Is angle A obtuse or acute? Ex. 969. The sides of a triangle are 4, 13, and 15. Find the projection of 13 upon 4. Ex. 970. If the value of p obtained from the above formula equals zero, what does this result signify? Ex. 971. In ▲ abc, a = 15, b = 13, c = 14. Find he. Ex. 972. In ▲ abc, a = 17, b = 10, c9. Find he 335. COR. 1. In Aabc, if p denotes the projection of b upon c, p b2 + c2 - a2 336. COR. 2. If h, denotes the altitude upon c, = (a+b+c) (b + c − a) (a - - b + c) (a + b −c) ̧ 4 c2 Let a+b+c=2s, i.e. let s denote half the perimeter. Ex. 974. The three sides of a triangle are 4, 13, and 15. altitude upon 4. Find the Ex. 975. The three sides of a triang.e are 25, 30, and 11. Find the altitude upon 11. [See practical problems 80 to 83, p. 299.] PROPOSITION XXXVIII. THEOREM 337. In any triangle, the square of one side plus four times the square of the corresponding median is equal to twice the sum of the squares of the other sides. m D/ཤྭE Given in ▲ ABC, m, the median to c. c2+4 m2 = 2 a2+2 b2. Proof. Draw CEL AB, and suppose E to fall between 4 and To prove Ex. 976. The sides of a triangle are 7, 8, and 9 respectively. Find the length of the median to 8. Ex. 977. The sides of a triangle are 7, 4, and 9 respectively. Find the length of the median to 9. Ex 978. The sides of a triangle are 10, 5, and 9 respectively, Find the length of the median to 9. Ex. 979. The sides of a triangle are 22, 20, and 18 respectively. Find the length of the median to 18. Ex. 980. In ▲ abc, if m., denotes the median drawn to c, prove that mc= — √2a2 +262 — c2. |