Difficult plane geometry question (1 Viewer)

[c_james]

Any help with this one would be much appreciated:

 

[Winston]

Is any other information given for the question?

 

[c_james]

No. I'm quite sure it can be done, as my teacher went through it in class (I didn't get a chance to copy down his solutions though, as the bell rang).

 

[Winston]

Well seeing the Perp Bisector is BE, then it cuts that triangle up, so that BEA = BEC = 90 Degrees, and the other four remaining angles are respectively equal to 45 Degrees, and then and to get ACD you just do 180 - 45 degrees, since it's on a straight line.

so ACD = 135 Degrees, and we know that BAC = 45 Degrees, so add them together and you get 180 degrees. and 2 x BEC = 180 so yeah.

I don't think i've proved it right lol, cause i assumed the two triangles remaining angles are cut up evenly to 45 degrees.

 

[untamedanimal]

If you are given that AB = BC the Winston's solution would be correct, otherwise i dont think its provable

 

[...]

u dunno if BE cuts AC perpendicularly, thats the problem, it looks like it is, but theres no info stating it is...

 

[c_james]

Even if the two triangles were cut up evenly to 45 degrees, that doesn't necessarily mean AB will equal BC. My teacher did it using a bunch of exterior angles theorems, and I remember it was quite a long process. But I'm sure it's provable.

 

[Winston]

But it would be very hard, i mean i'm pretty sure there should be at least one piece of data, and the crucial item would be knowing that BE is perpendicular.

 

[c_james]

If only I'd got up off my lazy ass and copied the answers...I'm looking at the paper right now and no other information was provided. I remember my teacher let one of the top angles equal x and used a lot of algebra and such to reach the final answer.

 

[Affinity]

You don't need any additional assumptions.

ACD = CBE + BEC (exterior angle = sum of interior angles)

BEC = EBA + BAC (same reason)

now note that CBE = EBA, subtract the second equation from the first.

ACD - BEC = BEC - BAC
ACD + BAC = 2BEC

moral of the story: Don't dismiss something until you find some concrete counterexample

 

[elizabethy]

Originally posted by Affinity

moral of the story: Don't dismiss something until you find some concrete counterexample

this is so true!!