Номер 1149 — ГДЗ Геометрия 9 класс Атанасян

Дано:

$△ A BC$ , $a = BC$ , $b = A C$ , $c = A B$
а) $c = 8, ∠ A = 3 0^{\circ}, ∠ B = 4 5^{\circ}$
б) $c = 5, ∠ B = 4 5^{\circ}, ∠ C = 6 0^{\circ}$
в) $c = 3, a = 3, 3, ∠ A = 4 8^{\circ} 3 0^{'}$
г) $b = 10, 4, a = 5, 2, ∠ B = 6 2^{\circ} 4 8^{'}$

Найти:

все стороны и углы $△ A BC$

Чертёж к задаче 1149 — треугольник ABC со сторонами a, b, c

Решение:

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

а) $c = 8, ∠ A = 3 0^{\circ}, ∠ B = 4 5^{\circ}$

∠ C = 18 0^{\circ} - (3 0^{\circ} + 4 5^{\circ}) = 10 5^{\circ}

a = \frac{c sin A}{sin C} = \frac{8 \cdot sin 3 0 ^{\circ}}{sin 10 5 ^{\circ}} \approx \frac{8 \cdot 0 , 5}{0 , 9659} \approx 4, 14

b = \frac{c sin B}{sin C} = \frac{8 \cdot sin 4 5 ^{\circ}}{sin 10 5 ^{\circ}} \approx \frac{8 \cdot 0 , 7071}{0 , 9659} \approx 5, 86

б) $c = 5, ∠ B = 4 5^{\circ}, ∠ C = 6 0^{\circ}$

∠ A = 18 0^{\circ} - (4 5^{\circ} + 6 0^{\circ}) = 7 5^{\circ}

a = \frac{c sin A}{sin C} = \frac{5 \cdot sin 7 5 ^{\circ}}{sin 6 0 ^{\circ}} \approx \frac{5 \cdot 0 , 9659}{0 , 8660} \approx 5, 58

b = \frac{c sin B}{sin C} = \frac{5 \cdot sin 4 5 ^{\circ}}{sin 6 0 ^{\circ}} \approx \frac{5 \cdot 0 , 7071}{0 , 8660} \approx 4, 08

в) $c = 3, a = 3, 3, ∠ A = 4 8^{\circ} 3 0^{'}$

sin C = \frac{c sin A}{a} = \frac{3 \cdot sin 4 8 ^{\circ} 3 0 ^{'}}{3 , 3} \approx \frac{3 \cdot 0 , 7490}{3 , 3} \approx 0, 6809

∠ C \approx 4 2^{\circ} 5 5^{'}

∠ B = 18 0^{\circ} - (4 8^{\circ} 3 0^{'} + 4 2^{\circ} 5 5^{'}) = 8 8^{\circ} 3 5^{'}

b = \frac{a sin B}{sin A} = \frac{3 , 3 \cdot sin 8 8 ^{\circ} 3 5 ^{'}}{sin 4 8 ^{\circ} 3 0 ^{'}} \approx \frac{3 , 3 \cdot 0 , 9997}{0 , 7490} \approx 4, 40

г) $b = 10, 4, a = 5, 2, ∠ B = 6 2^{\circ} 4 8^{'}$

sin A = \frac{a sin B}{b} = \frac{5 , 2 \cdot sin 6 2 ^{\circ} 4 8 ^{'}}{10 , 4} = \frac{1}{2} sin 6 2^{\circ} 4 8^{'} \approx \frac{0 , 8894}{2} = 0, 4447

∠ A \approx 2 6^{\circ} 2 4^{'}

∠ C = 18 0^{\circ} - (6 2^{\circ} 4 8^{'} + 2 6^{\circ} 2 4^{'}) = 9 0^{\circ} 4 8^{'}

c = \frac{b sin C}{sin B} = \frac{10 , 4 \cdot sin 9 0 ^{\circ} 4 8 ^{'}}{sin 6 2 ^{\circ} 4 8 ^{'}} \approx \frac{10 , 4 \cdot 0 , 9999}{0 , 8894} \approx 11, 69

Ответ: а) $∠ C = 10 5^{\circ}, a \approx 4, 14, b \approx 5, 86$ ; б) $∠ A = 7 5^{\circ}, a \approx 5, 58, b \approx 4, 08$ ; в) $∠ C \approx 4 2^{\circ} 5 5^{'}, ∠ B = 8 8^{\circ} 3 5^{'}, b \approx 4, 40$ ; г) $∠ A \approx 2 6^{\circ} 2 4^{'}, ∠ C = 9 0^{\circ} 4 8^{'}, c \approx 11, 69$ .

Номер 1149 — ГДЗ Геометрия 9 класс Атанасян

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