I don't believe there is enough information to answer this question. The reason being because we do not know which sides correspond between the two triangles. We need to know this so that we can determine the scaling factor.
Question 4 on this piece of paper that is his homework reads...
Triangle PQR ~ Triangle DEF.
Determine the length of DF
If PQ = 10cm, DE=5cm and PR=6cm.
Now a search of the net thank you Mr Google lets me know the the tilda sign ~ appears to mean "Similar"
And two triangles are similar if they gave the same corresponding angles but can have different sizes and rotation.
I suspect that just because someone writes
"Triangle PQR ~ Triangle DEF"
As I understand it, this does not mean that PQ actually equals DF and so on.
So depending on the rotation or definition of DEF, Then DF could be 3 or 8.3333 or something else.
So it seems to me that we need more info.
The challenge is over to you. Can someone answer this question or tell me definitively that it cant be answered.
Thanks.
I don't believe there is enough information to answer this question. The reason being because we do not know which sides correspond between the two triangles. We need to know this so that we can determine the scaling factor.
I don't see the problem?
The same result can be achieved by stating
EDIT: I see now why this can be complicated. Of course, without knowing what sides the teacher assigns the letters to, the question remains ambiguous. Sorry for my intrusion.
When we say that triangle PQR is similar to triangle DEF it means that point P corresponds to point D, Q to E and R to F.
Thus we have PQ/DE=QR/EF = PR/DF
Now substitute the given information and complete the question.
In fact that is the truth. we have to mention the vertices of similar / congruent triangles in the same sequence.
pl visit the site. https://en.wikipedia.org/wiki/Similarity_(geometry)
There is nothing in that article which states what you are claiming, and even if there was, Wikipedia is not a reliable resource. We do not have to mention the vertices in the same sequence. I admit, it would be easier if we did, but it is not a convention as you are claiming.
Thanks guys for the replies .....
I suspect that just because someone writes
"Triangle PQR ~ Triangle DEF"
this does not mean that PQ actually equals DF and so on.
So depending on the rotation or definition of DEF, Then DF could be 3 or 8.3333 or something else.
Don't know if this helps, but if I was given that I'd be expected to consider all possible cases unless it's formally defined. If the teacher has not given a definition of ~, then it can be answered depending on how complicated you want to make it and how many assumptions you wish to make. For instance, since this is grade 9, it is reasonable to assume that this is over the Euclidean plane and that you cannot have an angle of 0 degrees. Then side QR has length 4<x<=16, and there will only be 6 cases: (also keeping your intuitive meaning of ~, forgetting symmetry and bearing in mind my arithmetic is rubbish)
1) PQ ~ DE and PR ~ DF, leading to DF having length 3.
2) PQ ~ DE and QR ~ DF, leading to DF having length 2<x<=8
3) PQ ~ DF and PR ~ DE, leading to DF having length 8,1/3
4) PQ ~ DF and QR ~ DE, leading to DF having length 3.125<=x<12.5
5) PQ ~ EF, so PR ~ DE and QR ~ DF, leading to DF having length 3,1/3<x<=13,1/3
6) PQ ~ EF, so PR ~ DF and QR ~ DE, leading to DF having length 1.875<=x<7.5
(I hope my arithmetic was correct, been a long time since I've worked with actual numbers)
So, the answer would be: As you have not defined ~ sufficiently, I've made some basic assumptions leading to these as your possible answers. If you did not mean any of these, please define your terms and give me the question again.
That is correct
You can disagree if you want. But it does change the fact that has been the convention understood since David Hilbert gave the modern version of the axiomatic geometry. Now I admit that low level mathematics courses were very slow to adopt this rigor. The value in the notation clear in this very problem. There is no need to list out the correspondence.
Plato, surely that would depend on how far you are in the course? If you are speaking German to someone who is German, then you adopt their sentence structure as it is the convention. If you are learning German for the first time, then you would not be aware of the convention unless it is defined. Plus, although a symbol may have a conventional meaning, there is nothing to stop it being used for another purpose. Ergo, even if that is the convention, there is no reason to believe that the teacher uses it that way? (Unless of course it is widely used, like the word banana. But in this case, I use ~ in many different contexts, not one to do with geometry.)
It is not a matter of language. It is a matter of mathematics. Have you studied axiomatic geometry?
I know of no college level geometry textbook that does not use that convention.
No one teaching a geometry course has any excuse not to know this cvonvetion.
After all it is over 100 years old.
I understand that - I am not debating over whether it is the convention. I'm merely trying to say that if you are introduced to this in passing, it may not have been mentioned that this is the convention. Although the teacher knows to follow this, to a student it may not be clear. In that case, all the student can do is consider all possible cases until he finds out for certain. Hence, if he just wants to answer the question, surely it is better to consider all possible cases and lose a few marks for not knowing what ~ means rather than resort to learning axiomatic geometry, especially if you don't know that that is where to look? For reference, I have studied it, yes.
Yes, you should follow convention and, here, the solution is obvious. However, if you don't know and you come across a question like this again, where there is something you don't know, maybe it would be nice to realise you can still tackle the problem.
At the cost of repetition I would like to mention that it is not a matter of convention. It is required that we must write the triangles involved with the vertices having equal angle in that order: For example in triangles ABC and DEF if we have angle A = angle F; angle B = angle E; and angle C = angle D then the correspondence of vertices would be: A to F; B to E and C to D. In this case we have to write triangle ABC is similar to triangle FED. In case the order is not retained it would be a wrong statement. Following are wrong statements:
triangle ABC is similar to triangle EDF
triangle ABC is similar to triangle DFE
triangle ABC is similar to triangle DEF
triangle ABC is similar to triangle EFD
triangle ABC is similar to triangle FDE
And the following are all correct
triangle BCA is similar to triangle EDF
triangle CAB is similar to triangle DFE
I hope now the concept would be clear