Need help with Geometry and Ratio Question!! Please Help!!

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Need help with Geometry and Ratio Question!! Please Help!!

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The base of a triangular piece of paper ABC is 12 cm long. The paper is folded down over the base, with crease DE parallel to the base of the paper. The area of the triangle that projects below the base is 16% that of the area of the triangle ABC. What is the length of DE, in cm.
|dw:1327792772445:dw|
|dw:1327793675334:dw| ok so we actually are interested in the area of the triangle CFG too, since that is the one that projects below the base.

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Yes
What is the connection between those triangles?
These are similar triangles. There exists a scale factor s>1 such that \[ABC = s^2CFG.\] Another fact about s is that \[FGs = AB = 12\] Therefore \[s = \frac{12}{FG}.\] There also exists another scale factor, r>1 such that \[ABC=r^2CDE\] and obviously another fact about r is that \[rDE = AB = 12\] Therefore \[r=\frac{12}{DE}\] We are also told that the area of the triangle that projects below the base is 16% that of the area of the triangle ABC. So we get \[0.16ABC = CFG.\] What I think we should do is solve for r solve for s and then solve for DE.
i understand that they are similar but I don't quite understand why is (s^2)(CFG)=ABC. Why does s have to be squared?
is it some sort of formula?
\[0.16s^2CFG = CFG \Longrightarrow s=\sqrt{\frac{1}{0.16}} = 2.5\] Also, \[s=\frac{12}{FG} \Longrightarrow FG = 4.8\]
well similar lengths are scaled by a factor, and similar areas are scaled by the square of that factor
ok so we have found s, maybe try to find r next
oh, i see.
so does it work the same for r too?
|dw:1327795417846:dw|
r is a little more tricky...
because we don't know what percent is CDE of ABC
actually no... \[(0.4)^2 ABh_1 = FGh_2\] So it must be that \[0.4h_1 = h_2\]
yes, but what about CED?
Yeah, we need to try to figure out something relating s and CDE so we can use \[ABC = r^2CDE\]
oh, ok
I'm going to think about it for a bit lol
ok,thx
  • phi
If you want a hint. I hope this is not too cryptic
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Can you clarify ? |dw:1327797362393:dw|
why is it 2/5?
  • phi
0.4 you figured that out up top. It is the ratio of altitudes of 2 similar triangles. When the areas are in the ratio of 0.16/1 the altitudes are in the ratio sqrt(.16)/sqrt(1)= 0.4= 2/5
oh yeah. Thank you.
yeah phi that's excellent, 0.4 is the scale factor for the heights of the triangles I had above: \[0.4h_1 = h_2\] and yeah r is the ratio of the heights of CDE and ABC. Thanks phi
Thank you, guys!!!

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