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r - promień okręgu wpisanego = 4
IABI - dolna podstawa czworokąta = a
IEFI - górna podstawa czworokąta = b = 2r = 8
∡α = 60°
(a - b)/2 : r = ctgα = ctg60° = √3/3
3(a - b) = 2√3/3
(a - b) = 2√3/6
(a - 8) = 2√3/6
a = 2√3/6 + 8 = (2√3 + 48)/6 = 2(√3 + 24)6 = (√3 + 24)/3
P - pole czworokąta = [(√3 + 24)/3 + 2r) * r/2 = [(√3 + 24)/3 + 8] * 2 =
= [(√3 + 24 + 24)/3] * 2 = [(√3 + 48)/3] * 2 = 2/3 * (√3 + 48)
zad 8
∡ = 150°
naprzeciwległy bok = 4
twierdzenie sinusów
4/sin150° = R
4/sin(180° - 150°) = R
4/sin30° = R
4 : 1/2 = R
R - promień okręgu opisanego = 4 * 2 = 8
odp B
zad 9
α - ∡BAC
twierdzenie sinusów
√6/sin60° = 2/sinα
√6 : √3/2 = 2/sinα
√6 * 2/√3 = 2/sinα
2√2 = 2/sinα
2√2 sinα = 2
sinα = 2 : 2√2 = 1/√2 = √2/2
α = 45°
odp B
zad 10
twierdzenie cosinusów
IACI = b = ?
IABI = c = 3
IBCI = a = 2
α = 60°
b² = a² + c² - 2ac * cos60°
b² = 2² + 3² - 2 * 2 * 3 * 1/2 = 4 + 9 - 12 * 1/2 = 13 - 6 = 7
b = √7