中五數學題

This is an AS, GS question. Please draw a graph to help you understand the below steps.
[i]
Assume each side of the triangle p1q1r1 is X.
Then its permeter is 3x.
Then since it is an equi. triangle, its angle would be 60 degree and p1p2 = p1q2 [suppose p2 is the mid-pt of p1r1 and q2 is the mid-pt of p1q1], so triangle p1q2p2 is an isos. triangle.
As a result, the other two angles of it are also 60 degree.
Then it is proved that actually p1q2p2 is an equilateral triangle and p2q2 = x/2
As a result perimeter of p2q2r2 = 3x/2
the ratio = perimeter of p2q2r2 / perimeter of triangle p1q1r1 = 1/2 = 0,5

[ii]perimeter of triangle p1q1r1 =a
a(0.5)(0.5)(0.5)(0.5)(0.5)=1
so a =32cm

i) because triangle P2Q2R2 is constructed by joining the mid-points of every side of triangle P1Q1R1and P1Q1R1 is an equilateral triangle. So , triangle P2Q2R2 is similar to P1Q1R1. So as the situation of P3Q3R3.... P6Q6R6, they are all similar to P1Q1R1.

So that, when the perimeter of triangle P6Q6R6 is 1cm, the ratio is 1:1:1.
As a result ,the ratio between the perimeter of triangle P1Q1R1 and triangle P2Q2R2 are also 1:1:1.

ii) Because triangle P2Q2R2 is constructed by joining the mid-points of every side of triangle P1Q1R1 and they are also similar. So,
the perimeter of triangle of triangle P1Q1R1 = 1/3*2*2*2*2*2*3
= 64cm


P.S: 1/3 = a side of P6Q6R6
*2 = a side of P5Q5R5
*2 = a side of P4Q4R4
*2 = a side of P3Q3R3
*2 = a side of P2Q2R2
*2 = a side of P1Q1R1
*3 = trianglehas 3 sides

2007-10-19 23:05:32 補充：
有需要的話我可以換一下中文