1. Используя рисунок, выразите отрезки AС и АD через тригонометрические функции углов α и β [2]

2. Две стороны прямоугольного треугольника равны: 6 см и 8 см. Найдите третью сторону треугольника. Рассмотрите все возможные случаи. [3]

3. В прямоугольном треугольнике cos α=45. Вычислите sin α, tg α и ctg α. [4]

4. В равнобедренной трапеции ABCD сторона ВС равна 5 см, высота СЕ равна 3√3 , а боковая сторона образует с основанием АD угол 60°. Найти основание AD трапеции ABCD. [5]

5. Найти углы ромба ABCD, если его диагонали АС и BD равны 4√3 м и 4 м. ​

1. S = ½×(4+8)×5 = ½×6×5 = 3×5 = 15 см².

2. S=150, h=S:(½×(a+b)) = 150:(½×(9+11)) = 150:(½×20) = 150:10 = 15 см.

3. Пусть высота будет BH(нужно отметить Н на рисунке). Проведём высоту из точки С, будет она СЕ. Т.к. трапеция равнобедренная, то АН=DE. AH=BH=4 см, ведь угол А=45°, угол Н=90°, соответственно угол В=45° и треугольникк АВН равнобедренный. Из этого, AD=4+5+4 = 13 см.

Найдём площадь: S=½×(5+13)×4 = ½×18×4 = 9×4 = 36 см².

4. Пусть одна часть будет х, тогда BC=3x, AD=4x.

S=½×(3x+4x)×5 = ½×7x×5 = 3,5x×5 = 17,5x -> 17,5x = 35.

x=2 см.

AD=4x = 4×2 = 8 см.

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