Reading (5 points)
Task 1. Read the text:
Nowadays people have started to realize that good health is very important. much more important than
wealth. To be in good health people must follow a healthy life style to do things that will help keep a
healthy body and mind, to feel fit. There are several ways to do it. The state of your body depends on how
much time you spend doing sports. At least everybody must do moming exercises every day. The healthiest
kinds of sports are swimming, running and cycling. Healthy food is also a very important factor. The daily
menu should include meat, fruit and vegetables, milk product, which are rich in vitamins, fax, proteins and
etc. On the other hand modem diets are very popular especially among women. Diets may be harmful, if
they are used in the wrong way. To be healthy, people should get rid of their bad habits. It's necessary to stop
smoking and drinking much. Everyone should remember that citarettes, alcohol and drugs destroy both
body and brain. In addition it is recommended to watch TV less, avaid anxiety and observe daily routine.
Read the statements and choose T (True), F (False )or DS (doesn't say)
1)Good health is very significant.
2)All diets are useful and healthy.
3)It is recommended to do yoga to be in good health.
4) Alcohol and cigarettes influence only on body.
5)It is important to get rid of bad habits in order to keep a healthy body and mind.
Пусть H - середина ABCD, MH - высота пирамиды MABCD,
MH - медиана, биссектриса и высоты треугольника DBM => H - середина DB=> HL - средняя линия треугольника DMB => 2LH=DH;
AH перпендикулярно BD ( как диагонали квадрата),
AH перпендикулярно МH ( т.к. МH - высота пирамиды)
DB пересекает MH в точке H => AH перпендикулярна плоскости DMB, значит угол HLA = 60° (по условию),
CA = √(CB^2+AB^2)=6√2 (по теореме Пифагора)
HA=1/2CA=3√2
LM=AH/tg60° = √6
DM=2LM=2√6
MH=√(DM^2-DH^2)=√6 (по теореме Пифагора)
ответ: √6
О - центр окружности - точнее, обеих окружностей, заданных в задаче (ясно, что точки А1 В1 С1 равноудалены от центра вписанной окружности, то есть окружность, вписанная в АВС и окружность, описанная вокруг А1В1С1 - и проходящая через А - имеют общий центр).
В треугольнике АС1О стороны ОС1 и ОА равны, и - кроме того, медиана АВ перпендикулярна стороне ОС1. То есть АС1О - равносторонний треугольник.
Аналогично и АВ1О - равносторонний треугольник, но уже и без того ясно, что угол ВАО = 30 градусам, а угол САВ = 60 градусам.
Отсюда по теореме синусов 2Rsin(60°) = a; R = a/√3;