1. Choose the correct answer (10 min) 1. The boys…..,when their father came in.
a) fight b) fought c) fights d) were fighting e) are fighting
2. While Nelly……..to her cousin, I……a letter.
a) spoke/was writing b) was speaking/was writing c) was speaking/wrote
d) was speaking/write e) speak/was writing
3. My house is …..the shop and the theatre.
a) among b) long c) between d) along e) across
4. I like to be in a……air.
a) deep b) high c) tall d) fresh e) hide
5. Those flowers smell…….. .
a) pleasant b) young c) deep d) high e) tall
6. Breathe …! Breathe …. !
a) on/out b) at/out c) in/off d) in/out e) in/of
7. Antonym for the word “possible” is….. .
a) unpossible b) impossible c) dispossible d) mispossible e) inpossible
8. The book……everywhere.
a) are sold c) is sold d) is sell d) am sold e) are to sell
9. A lot of houses ……in the city every year.
a) is build b) are build c) am build d) is built e) are built
10. A ……..winter is not very cold and snowy.
a) fresh b) deep c) mild d) high e) tall
.
В нашем случае искомый угол - это угол между высотой СН треугольника (плоскости) АВС и высотой DH треугольника (плоскости) DAB.
Поместим начало координат в точку D(0;0;0). Тогда имеем точки:
А(0;а;0), В(0;0;а), С(а;0;0).
Найдем координаты точки Н, как середины отрезка АВ:
Н(0;а/2;а/2).
Тогда вектор DH{0;а/2;а/2}, его модуль |DH|=√(2a²/4)=a√2/2,
вектор СН{-a;a/2;a/2}, его модуль |HC|=√(6a²/4)=a√6/2.
Cosα=(x1*x2+y1*y2+z1*z2)/(|DH|*|HC|) или
Cosα=(0+а²/4+а²/4)/(а²√12/4)=(2а²*4)/(4*а²√12)=2/√12=√3/3.
ответ: Искомый угол равен α=arccos√3/3 или α≈54,74°.