Why Are There Two Possible Values of x in Similar Shapes Geometry Problems?

[TensorCalculus]

Also, as a tutor I very often work with materials that school/college teachers provided, and sometimes there is pure malice on their part. There were situations where students wrote to some "higher power" about that and I was asked to be a 'reviewer'. There was two teachers (one in high-school, the other one at my own uni, not related) who did not give maximum amount of points for a perfectly good answer, because it was not clever enough! Of course they had those very clever way of reasoning prepared, but please... These people should not teach. Fortunately "higher powers" did their job, and in both cases students got their points back.

It's a shame that happens, genuinely. And I agree that those people should not teach. But in this case, it's less about how clever the solution is and more about the problem itself: even if the intent was to malicious, the fact still holds that the OP is learning a valuable lesson here, through the struggle. And in this case, in my eyes that makes it okay to give students this question.

Maybe on the actual GCSE the diagram should have been more accurate (though they should not draw their diagrams to scale, ever) but in this case, it's a good thing the diagram is the way it is. The OP is learning and regardless of the intent of the writer, it's a good thing in the end.

 

[DaveC426913]

...the fact still holds that the OP is learning a valuable lesson here, through the struggle.

... they should not draw their diagrams to scale, ever) but in this case, it's a good thing the diagram is the way it is. The OP is learning and regardless of the intent of the writer, it's a good thing in the end.

My smack is gobbed that we here are (apparently) not all unanimous in this view.

 

[renormalize]

My smack is gobbed that we here are (apparently) not all unanimous in this view.

Are you sure it wasn't your gob that was smacked?

 

[fresh_42]

I am really surprised that some seriously expect to take BE and CD as parallels and don't have any problems taking 12 and 3 being different, although they look almost equal. That simply doesn't make any sense.

 

[DaveC426913]

Not to mention that the problem explicitly says there are two solutions. If the student finds one solution and stops there, they have not understood the assignment. That's a lesson unto itself.

So, there's no trick here, it's merely that it's not a dead-easy problem; it requires the student to do some thinking.

 

[votingmachine]

Homework Statement: See screenshot in main body
Relevant Equations: I don't know if there is a generic equation
Maybe something like AB(SF)=AC ?

View attachment 363169
Why are there two possible values of x?

Here's what I've done so far.

AE(SF)=AD
12(SF)=15
SF=15/12
=5/4

AB(SF)=AC
8(5/4)=AC
AC=10

AC-AB=x
x=10-8
x=2

A hint please, as to my next step

You are assuming that there is only one way the triangles can be similar ... the drawing leads you there ... I suspect you need to evaluate the words and make your own drawings.

 

[fresh_42]

Not to mention that the problem explicitly says there are two solutions. If the student finds one solution and stops there, they have not understood the assignment. That's a lesson unto itself.

So, there's no trick here, it's merely that it's not a dead-easy problem; it requires the student to do some thinking.

... plus, there is a symbol available in geometry to show parallelity in pictures. //

 

[DaveC426913]

... plus, there is a symbol available in geometry to show parallelity in pictures. //

You mean >

 

[mathwonk]

What surprised me was that there seem to be more than 2 ways the triangles can be similar, but they yield only 2 results for the question asked. Did this surprise anyone else? Or am I wrong?

 

[fresh_42]

You mean >

I know it with two short lines, but anyway. If the sides were parallel, there would have been a possibility to mark it accordingly.

 

[fresh_42]

What surprised me was that there seem to be more than 2 ways the triangles can be similar, but they yield only 2 results for the question asked. Did this surprise anyone else? Or am I wrong?

I thought that there were just two orientations possible, and that is the reason why, but I admit, I haven't considered this in detail.

 

[TensorCalculus]

What surprised me was that there seem to be more than 2 ways the triangles can be similar, but they yield only 2 results for the question asked. Did this surprise anyone else? Or am I wrong?

The fact that $\angle BAE = \angle CAD$ eliminates the other options, right?

 

[mathwonk]

the triangles could be isosceles, and hence have self similarities. right? i.e. we seem to be making the unstated assumption that the similarity must fix angle A. It seems angle BAE could equal angle BEA, no?

 

[TensorCalculus]

Well if the triangles are isosceles, then there would still only be 2 answers for x, no?

EDIT: I need to write this down on a piece of paper. I'm pretty much just guessing based on what would make sense but I haven't actually tried solving for x in the case where the triangle is isosceles

 

[mathwonk]

yes, I believe that is what I said.

 

[DaveC426913]

View attachment 363376

I know it with two short lines, but anyway. If the sides were parallel, there would have been a possibility to mark it accordingly.

I've not encountered that notation.

Where I come from, | means equal length. To-wit:

AB and DC are parallel.
AD and BC are of equal length.

 

[TensorCalculus]

yes, I believe that is what I said.

Oh, I thought you meant that the question only asked for 2 results but there are more than two orientations. My bad (you will find that I am pretty good at misinterpreting online messages ).

I've not encountered that notation.

Where I come from, | means equal length. To-wit:

AB and DC are parallel.
AD and BC are of equal length.

View attachment 363377

Same here in Britain (and since the OP seems to be preparing for GCSE this is the notation that they would see).
Though maybe @fresh_42 means not on the diagram but when referencing the lines. E.g. I have seen the notation $AB\parallel CD$ before.

 

[mathwonk]

I admit I still find this confusing, but it seems to me that all 6 possible similarities can occur, still with only two final answers of course.
I mention this because it seems to me the question can be viewed as having 2 parts: 1) how many different ways can the triangles be similar, and then 2) how many different values for x do these give? Since the lesson is about not making unjustified assumptions, I did not want to assume the similarity must fix A.

 

[DaveC426913]

Well if the triangles are isosceles, then there would still only be 2 answers for x, no?

EDIT: I need to write this down on a piece of paper. I'm pretty much just guessing based on what would make sense but I haven't actually tried solving for x in the case where the triangle is isosceles

Since the two triangles are defined as similar, there must be two possible solutions (since they share an angle in the diagram).

If those two triangles happen to be isosceles, there may be more solutions, but we cannot say they are isosceles; we can only speculate that they might be. So, we can speculate that there might be more solutions conditionally. But this is well beyond the scope of the question.

 

[DaveC426913]

I admit I still find this confusing, but it seems to me that all 6 possible similarities can occur, still with only two final answers of course.

Do you mean could occur if all numbers were put to the diagram?

Because we can't just pretend, say, that the triangles are isosceles, let alone equilateral.

The diagram is fixed by what is labeled. If it's not labeled, it can't be assumed or even speculated.

Let me illustrate that:

BE is not equal in length to AE.
Even if BE were measured to be exactly 12cm, it is still not geometrically the same length as AE.

Geometrically, there are only two ways to determine that two lines are equal: by definition, or by proof.

The problem/diagram provides neither, therefore no valid solutions can depend on that condition.

 

[mathwonk]

I believe all 6 similarities are possible with the numbers as given. E.g. with the numbers as given, one can apparently have angles BAE, ABE, ADC, all equal. Then the "rotation" taking vertices A-->D, B-->A, E-->C, seems to be a similarity from triangle ABE, to triangle DAC. Does that seem right?

 

[hutchphd]

Geometry is not about accurate drawings and measurements; it is about rough diagrams, properly labeled, and never making assumptions based on a diagram.

I was not teaching a course in geometry, but engineering physics. Including a diagram that tries to misrepresent the discription seems , on its face, both unwise and unfair to the student.
I would not advocate teaching geometry without more rigor than "yep those look parallel to me........don't need no stinkin' postulate".
You also seem to misunderstand (or misrepresent?) some other things I said and would ask you to look at them again. I said them as well as I know how. Perhaps I should have included a drawing ?

 

[DaveC426913]

I was not teaching a course in geometry, but engineering physics. Including a diagram that tries to misrepresent the discription seems , on its face, both unwise and unfair to the student.

Totally 110% agree. Engineering is a completely different animal.

I would not advocate teaching geometry without more rigor than "yep those look parallel to me........don't need no stinkin' postulate".

Absolutely. In engineering, two lines can be parallel if they are measured to be parallel (to within some degree of tolerance).

You also seem to misunderstand (or misrepresent?) some other things I said

I know. I'm having trouble interpreting what you wrote. I did my best.

If I were to try to put my finger on why I'm having trouble, it would be because you're using passive voice a lot, which makes it difficult to tell what you are arguing:
"Thing X is done." (Is it good or bad that thing X is done? Which way do you want me to take that?)
"Do thing X." (OK, you're saying doing thing X is clearly the correct option.)

 

[hutchphd]

... plus, there is a symbol available in geometry to show parallelity in pictures. //

But using that symbol in the "official" drawing would be a lie.
It so happens that the assumption of || does lead to a valid solution: but it precludes others. Such is the usual nature of any "Ansatz" used for solution. That's the true wisdom here IMHO

 

[DaveC426913]

:furiously Googles "ansatz":

 

[Muu9]

I dislike problems like this where the figure is maliciously drawn way out of scale.

Given that there are two different values of x, it would be impossible to draw the triangles to scale in this problem, as if you drew them to scale for one value of x, it would appear "maliciously" drawn way out of scale for the other value of x.

 

[TensorCalculus]

Given that there are two different values of x, it would be impossible to draw the triangles to scale in this problem, as if you drew them to scale for one value of x, it would appear "maliciously" drawn way out of scale for the other value of x.

exactly! If you draw the diagram to present the two lines as not parallel, then it's out of scale for the solution x=2 where the lines would be parallel!

 

[fresh_42]

But using that symbol in the "official" drawing would be a lie.
It so happens that the assumption of || does lead to a valid solution: but it precludes others. Such is the usual nature of any "Ansatz" used for solution. That's the true wisdom here IMHO

I think this question is not about the intercept theorems, but rather about correctly analyzing the set of given conditions without adding assumptions that have not been made. To me, it is a lesson of not accepting only the obvious and thinking about the not-so-obvious. As I said, criminal investigators, but also scientists, and engineers have to deal with such situations on a daily basis. Where else should they learn this if not in their class that should teach thinking? Mathematics has been for a long time what we call a Geisteswissenschaft, and I think for a reason.

But I admit, I am against this in my mind, stubborn way of treating math classes as a place to learn algorithms instead of real math. If teachers were right, we wouldn't ultimately need any math classes any longer, since WA can do what kids learn at school; a depressing perspective.

 

[paulb203]

Thanks a lot guys.

I got there in the end, thanks to all of you, and a bit of extra help from Maths is Fun.

(Flip AEB on its head, as it were, to put E at the top, with A still at the bottom left, etc.

Other value of x = 14.5)

I found this one very tough, but then it was Higher tier (GCSE, UK), so it might be aimed at students who are hoping for an 8 or a 9. I put a lot of work in to get a 7 last year.

I’m now trying the A Level material (and revising GCSE as I go), but this was one of those questions that made me feel like giving up.

Onwards and upwards though.

Cheers.

P.S. I now know that congruent shapes are a special case of similar shapes - I think?! My brain is frazzled!
P.P.S. I also found out recently that a circle is a special kind of ellipse :)

 

[TensorCalculus]

Thanks a lot guys.

I got there in the end, thanks to all of you, and a bit of extra help from Maths is Fun.

(Flip AEB on its head, as it were, to put E at the top, with A still at the bottom left, etc.

Other value of x = 14.5)

I found this one very tough, but then it was Higher tier (GCSE, UK), so it might be aimed at students who are hoping for an 8 or a 9. I put a lot of work in to get a 7 last year.

I’m now trying the A Level material (and revising GCSE as I go), but this was one of those questions that made me feel like giving up.

Onwards and upwards though.

Cheers.

P.S. I now know that congruent shapes are a special case of similar shapes - I think?! My brain is frazzled!
P.P.S. I also found out recently that a circle is a special kind of ellipse :)

Maybe give L2 further maths a go first. It will set you up well for A level.