1) AC = 4, KC = KH = 3, AK = √(4^2 - 3^2) = √(16 - 9) = √7 cos C = cos H = 3/4 cos CKH = cos(180-C-H) = -cos(C+H) = -cos(2C) = 1 - 2*cos^2 C = 1 - 2*9/16 = -2/16 = -1/8 По теореме косинусов CH^2 = CK^2 + KH^2 - 2*CK*KH*cos CKH = 3^2 + 3^2 - 2*3*3(-1/8) = 9 + 9 + 9/4 = 81/4 CH = 9/2 = 4,5 BC = CH/cos C = (9/2):(3/4) = 9/2*4/3 = 6 S(ABC) = BC*AK/2 = 6*√7/2 = 3√7
2) MK = KB = 4, MB = 4√2 sin b = 3/4, cos b = √(1 - sin^2 b) = √(1 - 9/16) = √(7/16) = √7/4 PB = PA = MB*cos b = 4√2*√7/4 = √14 AB = CD = PB*√2 = √14*√2 = 2√7 tg b = sin b/cos b = (3/4):(√7/4) = 3/√7 = AD/CD AD = CD*tg b = 2√7*3/√7 = 6
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1) AC = 4, KC = KH = 3, AK = √(4^2 - 3^2) = √(16 - 9) = √7cos C = cos H = 3/4
cos CKH = cos(180-C-H) = -cos(C+H) = -cos(2C) = 1 - 2*cos^2 C = 1 - 2*9/16 = -2/16 = -1/8
По теореме косинусов
CH^2 = CK^2 + KH^2 - 2*CK*KH*cos CKH = 3^2 + 3^2 - 2*3*3(-1/8) = 9 + 9 + 9/4 = 81/4
CH = 9/2 = 4,5
BC = CH/cos C = (9/2):(3/4) = 9/2*4/3 = 6
S(ABC) = BC*AK/2 = 6*√7/2 = 3√7
2) MK = KB = 4, MB = 4√2
sin b = 3/4, cos b = √(1 - sin^2 b) = √(1 - 9/16) = √(7/16) = √7/4
PB = PA = MB*cos b = 4√2*√7/4 = √14
AB = CD = PB*√2 = √14*√2 = 2√7
tg b = sin b/cos b = (3/4):(√7/4) = 3/√7 = AD/CD
AD = CD*tg b = 2√7*3/√7 = 6