1) AC = √(AB^2 - BC^2) = √(10^2 - 8^2) = 6
S = AC*BC/2 = 6*8/2 = 24
2) S = 1/2*AB*BC*sin ∠ABC = 1/2*5*8*sin 60° = 20*√3/2 = 10√3
3) ∠DCA = 180° - ∠BCA = 180° - 120° = 60°
AD = CD*tg ∠DCA = 2*tg 60° = 2√3
S = AD*DB/2 = 2√3*5/2 = 5√3
4) BC = AB = 17
p = (AB + BC + AC)/2 = (17 + 17 + 16)/2 = 25
S = √[p(p-AB)(p-BC)(p-AC)] = √[25*(25-17)(25-17)(25-16)] = √(25*8*8*9) = 120
5) p = (AB + BC + AC)/2 = (15 + 26 + 37)/2 = 39
S = √[p(p-AB)(p-BC)(p-AC)] = √[39*(39-15)(39-26)(39-37)] = √(39*24*13*2) =
= √(3*13*4*6*13*2) = 13*2*6 = 156
6) AB = BC = AC/(2cos ∠CAB) = 6/(2cos 30°) = 6/(2√3/2) = 6√3/3 = 2√3
∠ABC = 180° - ∠CAB - ∠CBA = 180° - 30° - 30° = 120°
S = 1/2*AB*BC*sin ∠ABC = 1/2*2√3*2√3*sin 120° = 6√3/2 = 3√3
7) AD = CD = 6/√2 = 3√2
BD = AD/tg ∠ABD = AD/tg 30° = 3√2 / (1/√3) = 3√2*√3 = 3√6
S = BC*AD/2 = (3√2 + 3√6)*3√2/2 = 3√2*3√2/2 + 3√6*3√2/2 = 9 + 9√3
8) AB = BC = AC = OA*√3 = 4√3
S = AB^2*√3/4 = 16*3*√3/4 = 12√3
9) Проведем в треугольнике высоту BH
OH = √(OA^2 - AH^2) = √(5^2 - 4^2) = 3
BO = OA = 5
BH = BO + OH = 5 + 3 = 8
S = AC*BH/2 = 8*8/2 = 32
10) OA = OB = OC = 6; AB = OA + OB = 12
AC = AB*cos 15 = 12*√(2 + √3)/2 = 6√(2 + √3)
BC*sin 15 = 12*√(2 - √3)/2 = 6√(2 - √3)
S = AC*BC/2 = 6√(2 + √3)*6√(2 - √3) = 36*√(4 - 3) = 36
11) По одной гипотенузе я не могу найти площадь треугольника.
Если это треугольник (8, 15, 17) - единственный целочисленный треугольник с гипотенузой 17, то его площадь
S = AC*CB/2 = 15*8/2 = 60