311. In any triangle having an obtuse angle, the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides, plus twice the product of one of these sides and the projection of the other side upon it. с a C Hypothesis. In ▲ ABC, C is an obtuse Z. Conclusion. c2= a2+b2+2 a. på. Proof. 1. Draw AD 1 BC extended. Then Poc= CD=p. 2. Then in ▲ ABD, c2 = h2 + BD2. 3. But 4. = h2b2-p2; and BD = a+p. 312. Cor. If a, b, and c are the sides of a triangle: Why? Ex. 140. Is the greatest angle of the triangle whose sides are 8, 9, and 12 acute, right, or obtuse ? Ex. 141. Is the greatest angle of the triangle whose sides are 12, 35, and 37 acute, right, or obtuse? Ex. 142. Prove that the sum of the squares of the diagonals of a parallelogram equals the sum of the squares of the sides of the parallelogram. (Use § 310 and § 311.) Ex. 143. If AB and AC are the equal sides of an isosceles triangle, and BD is drawn perpendicular to AC, prove 2 AC × CD = BC2. Note. Supplementary Exercises 47 to 51, p. 293, can be studied now. 313. When the three sides of a triangle are known, the altitude to each side can be computed. 3. 4. 1. Assume AD = h ̧, and ≤ B to be an acute angle. 2. .. b2 = a2 + c2 −2 a · på or b2 = a2 + c2 — 2 ap. a+b+c=2 8. 10. [2 ac + a2 + c2-b22 ac [(a + c)2 — b2][b2 — (a — c)2] 4 a2 = 2 a (a+c+b)(a+c—b) (b + a −c) (b − a+c). 4 a2 ..a+b-c2s-2 c=2(s-c). Similarly, b+c-a=2(sa); and c+a-b = 2 (s — b). 11. .. hå 12. ... ha Similarly, and Ex. 144. Find the three altitudes of the triangle whose sides are 13, 14, and 15, getting the results correct to one decimal place. 314. In any triangle, the sum of the squares of two sides equals twice the square of half the third side plus twice the square of the median drawn to that side. Hypothesis. In ▲ ABC, CD is the median to side AB. Conclusion. a2+b2 = 2 2 +2 m2. Note. -ZADC is either a rt. 4, an acute Z, or an obtuse Z. When it is a rt. 4, the proof is quite easy. Proof. 1. Assume that ADC is obtuse and hence BDC is acute. 2. Draw CEL AB, so that DE=p. CD Suggestion. - Determine a2 from ▲ BCD by § 310; b2 from ▲ ACD by § 311; then add, so as to obtain a2 + b2. Note. - By Proposition XXIII, it is possible to determine the three medians of a triangle when the three sides of the triangle are known. 315. Cor. The difference between the squares of two sides of a triangle equals twice the product of the third side and the projection of the median upon that side. Suggestion.- Determine b2 and a2 and then subtract the value of a2 from that of b2. Ex. 145. Determine ma when b = 12, c = mine also my and m.. = 16, and a = 20. Deter Ex. 146. From the conclusion of § 314 derive a formula for me in terms of a, b, and c. PROPOSITION XXIV. THEOREM 316. In any triangle, the product of two sides equals the product of the diameter of the circumscribed circle and the altitude upon the third side. Hypothesis. A ABC is inscribed in O O; AD is a diameter Analysis and proof left to the pupil. See analysis of § 283. Hence, when the sides of a triangle are known, the diameter of the circumscribed circle can be computed. Ex. 147. Determine the diameter of the circle circumscribed about the triangle whose sides are 13, 14, and 15. Ex. 148. If two adjacent sides and one of the diagonals of a parallelogram are 7, 9, and 8 respectively, find the other diagonal. (§ 314.) Ex. 149. The sides AB and AC of ▲ ABC are 16 and 9 respectively, and the length of the median drawn from C is 11. Find side BC. (§ 314.) Ex. 150. If the sides of AABC are 10, 14, and 16, find the lengths of the three medians. Determine also the diameter of the circumscribed circle. PROPOSITION XXV. THEOREM 318. In any triangle, the product of any two sides is equal to the product of the segments of the third side formed by the bisector of the opposite angle, plus the square of the bisector. 4 of ▲ ABC, meeting BC at D. (Let BD=r, and DC= p.) Hypothesis. AD bisects Proof. 1. Circumscribe a circle about ▲ ABC. Note. This proposition makes it possible to compute the bisectors of the three angles of a triangle when the three sides of the triangle are known. Ex. 151. If c = 4, b = 5, and a = = 6, find tɅ. Suggestions.-1. It is necessary to find r and p first. This may be done by using § 270. r = 4 6-r 5 2. Then substitute in the conclusion of § 318. Note. Supplementary Exercises 52 to 56, p. 294, can be studied now. |