(初三)圓問題兩問，勁急!!!

1.
Let M be the point on AB such that O'M is perpendicular to AB.
Let radius of bigger circle (centre O) = b
Let radius of smaller circle (centre O') = a
OO' = a + b
OM = OA - OC = b - a
By Pythagoras theorem
O'M^2 + OM^2 = O'O^2
O'M^2 = (a + b)^2 - (b - a)^2 = 4ab...................(1)
Again by Pythagoras theorem
O'M^2 + BM^2 = O'B^2
BM = b + ( b - a) = 2b - a
so 4ab + (2b -a)^2 = O'B^2
O'B^2 = 4ab + 4b^2 + a^2 - 4ab = 4b^2 + a^2.....(2)
Again by Pythagoras theorem
O'B^2 = O'D^2 + BD^2
O'D = radius of smaller circle = a
so from (2) we get
4b^2 + a^2 = a^2 + BD^2
BD^2 = 4b^2
BD = 2b = AB.

2.
Let O be centre of the arc somewhere down below PN.
Let radius of circle = r
so OA = OB = OA' = OB' = OP = r.
Let M is the point on AB such that PNM is perpendicular to AB, so AM = MB = 60/2 = 30m.
Since PM = 18, so OM = r - 18
By Pythagoras theorem
AM^2 + OM^2 = OA^2
(30)^2 + (r - 18)^2 = r^2
900 + r^2 - 36r + 324 = r^2
36r = 900 + 324 = 1224
r = 34.
Now PN = 4, so ON = r - 4 = 34 - 4 = 30
Again by Pythagoras theorem
A'N^2 + ON^2 = OA'^2
A'N^2 + 30^2 = r^2 = 34^2
A'N^2 = 34^2 - 30^2 = 256
A'N = sqrt 256 = 16
so A'B' = 16 x 2 = 32m which is more than 30m,
so NO action should be taken when PN is 4m.




2009-07-25 18:30:25 補充：
Correction: Q1 Line 5 should be O'C, not OC. (O'C = a)

2009-07-25 19:38:29 補充：
To: myisland8132. I don't think the tangent from B to D is also a tangent to the larger circle. BD is not the same as AC.

2009-07-25 20:12:41 補充：
For Q1, O'D is perpendicular to BD, this is the property of tangent to circle, I am not sure if you learn this in F.3

1 由P延長交AC和BD於M,N
則AO=BO,BO=BP,BN=PN (切線性質)
又角OPN=角OBN=90
因此OBNP是正方形 (一组邻边相等且有一个角是直角的四邊形是正方形)。
因此OB=BN。同理可得DN=PN=BN=OB=OA
因此AB=OA+OB=DN+BN=BD。即自B點至⊙O1的切線長等於AB
2 AM=60=>OA=OB=OP=30=>ON=30-4=26 m
A'N^2=OA'^2-ON^2=30^2-26^2=224
A'N=149666=>AB=2A'N=29.9333 m
因此不需採取緊急措施

2009-07-25 18:38:16 補充：
第二條做錯了。

設M是AB中點,O是圓心。

2009-07-25 18:46:39 補充：
P穿過M.O到另一點Q

MA=MB=30,MP=18.MA^2=(PM*QM)

30^2=18*QM=>QM=50=>圓的半徑是(50+18)/2=34m

PN=4=>ON=34-4=30m

因此A'N^2=OA'^2-ON^2=34^2-30^2=256

A'B'=16*2=32m

因此不需採取緊急措施

2009-07-25 18:48:45 補充：
第2題糾正

2009-07-25 18:51:14 補充：
設AB中心M﹐P過M與圓心O到另一點Q

AM*BM=PM*QM=>QM=30^2/18=50
因此圓的直徑是50+18=68m﹐由PN=4=>PM=34-4=30m
因此A'N'=開方[OA'^2-ON^2]=開方[34^2-30^2]=16m
A'B'=32m=>不需採取緊急措施

2009-07-25 18:52:18 補充：
第2題糾正

設AB中心M﹐P過M與圓心O到另一點Q

AM*BM=PM*QM=>QM=30^2/18=50
因此圓的直徑是50+18=68m﹐由PN=4=>PM=34-4=30m
因此A'N'=開方[OA'^2-ON^2]=開方[34^2-30^2]=16m
A'B'=32m=>不需採取緊急措施