Tiêu đề 42 Area of a Triangle.pdf ✅

MSTC 42: Area of a Triangle For this discussion, consider a triangle ABC: Also, let the area be denoted as K to prevent confusion with the angle A. 1. Fundamental Formula The area of a triangle is most well-known as K = base × height 2 Where the “height” is an altitude to a “base”. 2. One angle given [DERIVATION] Using trigonometry, solve for the height to any side, say b height = c sin A Therefore, the area is K = 1 2 (base)(height) K = 1 2 b(c sinA) K = 1 2 bc sinA
Similarly, for other angles, K = 1 2 ab sin C K = 1 2 ca sin B 3. One side given [DERIVATION] Using the sine law, a sin A = b sin B = c sin C If the given side is a, then b = a sinB sin A Therefore, express the area as K = 1 2 ab sin C K = 1 2 a ( a sin B sinA ) sin C K = a 2 sinB sin C 2 sin A Similarly, on the other sides, K = b 2 sin C sin A 2 sin B K = c 2 sin A sin B 2 sin C 4. Three sides given [DERIVATION] From the formula K = 1 2 ab sin C Use cosine law, cos C = a 2 + b 2 − c 2 2ab From Pythagorean identities, sin C = √1 − cos2 C sin C = √1 − ( a 2 + b 2 − c 2 2ab ) 2
sin C = √ (2ab + a 2 + b 2 − c 2)(2ab − a 2 − b 2 + c 2) 4a 2b 2 Substitute to the equation, K = 1 2 ab√ (2ab + a 2 + b 2 − c 2)(2ab − a 2 − b 2 + c 2) 4a 2b 2 K = 1 4 √[(a + b) 2 − c 2][c 2 − (a − b) 2] K = 1 4 √(a + b + c)(a + b − c)(a − b + c)(−a + b + c) In terms of its semi-perimeter, s = a+b+c 2 , it results in Heron’s formula, K = √s(s − a)(s − b)(s − c) 5. Miscellaneous Formulas These may be additional formulas since some quantities are related to secondary parts of a triangle. All of these formulas resemble the Heron’s formula. 5.1. Given three medians Given the medians to the sides of a triangle ma, mb, mc , then the area of the triangle is K = 4 3 √m(m − ma )(m − mb )(m − mc ) Where m = ma + mb + mc 2 5.2. Given three altitudes Given the altitudes ha, hb, hc , then the area of the triangle is 1 K = 4√h (h − 1 ha ) (h − 1 hb ) (h − 1 hc ) Where h = 1 ha + 1 hb + 1 hc 2 5.3. Given three angles If the diameter of the circumcenter is D, then K = D 2√S(S − sin A)(S − sinB)(S − sin C) Where S = sin A + sin B + sin C 2