In the figure above, if AB is parallel to DE and DE = 2, what is the

Beautiful problem, Bunuel! (It´s hard to "open our mind" to general geometric positions!)

Our method does NOT recommend, in general, to modify the position of figures given BUT... in this case we believe it helped a LOT...

\(? = {\text{perim}}\left( {CDE} \right)\)

Let´s go straight to (1+2) to prove, using GEOMETRIC BIFURCATION, that the official answer (E) is indeed the right one!

Important:

01. Figures were created not in scale, but in a manner that (we believe) are "visually obviously constructible" to guarantee the VIABILITY of the bifurcation!

02. \({{BE} \over {BC}} = {5 \over 7}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{{EC} \over {BC}} = {2 \over 7}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\Delta {\rm{s}}\,\,{\rm{ratio}}\,\,{\rm{of}}\,\,{\rm{similarity}}\,\, = \,\,2:7\,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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I received a PM requesting that I post a solution.

Triangles that have two angle measurements in common are SIMILAR.
Because AB and DE are parallel:
angle CAB = angle CDE
angle CBA = angle CED
Since triangles ABC and CDE have two angle measurements in common, the two triangles are similar.
When triangles are similar, corresponding sides are in the SAME RATIO.

To make the math easier, let Statement 2 indicate that AC=14 instead of AC=13:



Statements combined:

Case 1:

Here:
BE=10 and BC=14, so BE is \(\frac{5}{7}\) of BC
AC=14 and AB=7
Corresponding sides are in the same ratio:
\(\frac{DE}{AB}= \frac{2}{7}\)
\(\frac{CD}{AC} = \frac{4}{14} = \frac{2}{7}\)
\(\frac{CE}{BC} = \frac{4}{14} = \frac{2}{7}\)
Perimeter of triangle DEC = 4+2+4 = 10

Case 2:

Here:
BE=10 and BC=14, so BE is \(\frac{5}{7}\) of BC
AC=14 and AB=7
Corresponding sides are in the same ratio:
\(\frac{DE}{AB}= \frac{2}{7}\)
\(\frac{CD}{AC} = \frac{4}{14} = \frac{2}{7}\)
\(\frac{CE}{BC} = \frac{5}{17.5} = \frac{10}{35} = \frac{2}{7}\)
Perimeter of triangle DEC = 4+2+5 = 11

Since the perimeter of triangle DEC can be different values, the two statements combined are INSUFFICIENT.


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We have got one pair of parallel sides and one common point C. Therefore the triangles should be similar and the sides will have the same ratio.
Statement 1 alone is insufficient as it only gives us the ratio between BE and BC.
Statement 2 alone is insufficient as we can calculate DC but nothing about BC.
Combining the two statements, we can calculate the value of BC and hence the overall perimeter of DEC. Hence Option C is the correct answer.

Please correct me if my reasoning is wrong.

Hi,
As the position of D and E is unknown ,therefore, we can't say that triangles are similar and ratio of sides are equal .
Hence measure of side BC can't be determined

Hence ans (E)
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PKN-
It is given that AB is parallel to DE, so Angle BAC=Angle EDC & Angle ABC = Angle DEC also Angle DCE is common in both the triangles. So Triangle ABC should be similar to Triangle DEC by AAA similarity rule. Please correct if i am wrong.

Yes ,your reasoning is absolutely correct.

But what do you want to justify?

I'm not sure, in case u want to say we can determine perimeter by combing both statements using the triangle similarity property (ratio of sides are equal) then scale factor would play a vital role.Scale factor is not unique.Therefore,the sides of the triangle are not unique.Hence perimeter is not unique.

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So this is my approach:

DE is given to us as 2.
Statement 1: BE is 5/7 of BC. Great. We have the ratio of sides and eventually the ~ triangles. Triangles are similar because 2 sides are parallel. One should be able to spot that. Apart from that you cannot find anything else.

Statement 2: AC is 13 and AB is 7. Firstly, I really don't need AB. So let's just ignore that. AC is 13. So we can split that in 2:5 ratio and find the sides Great. But what about BC? Is the triangle equilateral? Is it isosceles? I don't think so and we have no information to deduce that. So clearly not sufficient.

Combined, Well, doesn't really help.

IMO E.