That would help a lot in the coming exam. Thank you for imparting that knowledge!
Hm.. I don't know about that, but I once had a drill exercise about those and the the class came to this conclusion when it concerns similar polygons.
That very principle also applies to volumes.
Say 2 circles of radius 2 and 5 respectively. The ratio of length is 2:5, that for the areas is 4:25 and that for the volumes is 8:125
Okay, to come to my solution about it, I used variables instead of giving specific values.
I let HM = x, this means that AH = 2x (due to similar triangles and I know that 2AM = AB) and using Pythagoras, I get AM =
From this, we get AB = and HB = 4x.
From there, I can get angles BAH and HAP which are respectively and respectively.
In degrees those give 63.43 and 26.57 degrees (I kept all the other numbers to retain the accuracy as much as possible).
And then I get angles ABP = 45, ACP = 45, APB = 180-(45+63.43) = 71.57, APC = 180-(45+26.57) = 108.43 degrees.
Since the sides don't matter now, I put AB = 2 cm and AH = 1 cm. And from those and using the angles I just found, using the sine rule, I get side BP = 2.12 cm and PC = 0.943 cm. The ratio then turns out to be 9:4
Hello again gundanium,
Here is my geometry-algebra method
AB =6 ( any number could be used)
From P draw a perpendicular to AC meeting it @T
BC=6rad2 BP=y PC =6rad2-y PT=6rad2-y/2
TC =( 6rad2-y/2) AT =6-(6rad2-y/2)
AH =b BH =a both b and a are numbers found in my stage 1 post(after calcs)
Write two equations in x and y and solve for x and y
Hm... yes, and from your method, I found yet a shorter method!
One slight thing I saw, is that PT =
Anyway, triangles ABC and CPT are similar, meaning that the ratio BP:PC = ratio of AT:TC
Once you know TC (and you alreayd know AC), you can find AT and then the ratio.
You agree that angle PTC is 90 degrees and hence, triangle PTC is another isoseles right angled triangle, similar to triangle ABC.
From that, we know the length of PC, the length of PT = TC = PC/rad2 which is what you got and I corrected in my post.
And from there, yu use another ratio trick to get the ratio of BP:PC