The length of arc AXB is twice the length of arc BZC

[ghong14]

According to the question, the ratio of the lengths of arc AYC to arc AXB to arc BZC is 6: 2: 1

I understand from the question that ABC is twice BZC. Therefore the Ratio between AXB:BZC is 2:1. In addition, AYC is three times the of the length of AXB. The question I have is how is the ratio of AYC:AXB:BZC deduced?

I have having a hard time finding ratios when there are more than one relationship. For example in this problem both BZC and AYC share a common relationship with AXB. How do I calculate the final ratio between all three segments with this information?

A simpler version would be if ration of A:B is 1:2 and A:C is 1:3 what is the ratio of A:B:C?

Thanks!

[LazyNK]

Hi Ghong14,

The method to get the combined ratio is to scale (which could be upscale or downscale) one or both the ratio's provided in such a way that the common element in the two given ratio's becomes "same number". Once that is done, the combined ratio is simply the three numbers taken in respective order.

For the arc example, you have already done a reverse deduction as follows :
AXB:BZC=2:1 ---- equation-1
AYC:AXB=3:1 ---- equation-2
The common element in both ratios is AXB, but the number corresponding to AXB is "2" in equation-1 but "1" in equation 2. To make number for AXB same for both equations, multiply equation-2 by 2 on both numerator and denominator :
AYC:AXB=3:1=3*2/1*2=6:2 ---- equation-3
Now, using equation-1 and equation-3,
AYC:AXB:BZC=6:2:1

-NK

[tim]

Thanks Lazy. Ghong, let us know if you have any further questions..