anonymous
  • anonymous
Need help with Geometry and Ratio Question!! Please Help!!
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
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.
anonymous
  • anonymous
|dw:1327792772445:dw|
anonymous
  • anonymous
|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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anonymous
  • anonymous
Yes
anonymous
  • anonymous
What is the connection between those triangles?
anonymous
  • anonymous
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.
anonymous
  • anonymous
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?
anonymous
  • anonymous
is it some sort of formula?
anonymous
  • anonymous
\[0.16s^2CFG = CFG \Longrightarrow s=\sqrt{\frac{1}{0.16}} = 2.5\] Also, \[s=\frac{12}{FG} \Longrightarrow FG = 4.8\]
anonymous
  • anonymous
well similar lengths are scaled by a factor, and similar areas are scaled by the square of that factor
anonymous
  • anonymous
ok so we have found s, maybe try to find r next
anonymous
  • anonymous
oh, i see.
anonymous
  • anonymous
so does it work the same for r too?
anonymous
  • anonymous
|dw:1327795417846:dw|
anonymous
  • anonymous
r is a little more tricky...
anonymous
  • anonymous
because we don't know what percent is CDE of ABC
anonymous
  • anonymous
actually no... \[(0.4)^2 ABh_1 = FGh_2\] So it must be that \[0.4h_1 = h_2\]
anonymous
  • anonymous
yes, but what about CED?
anonymous
  • anonymous
Yeah, we need to try to figure out something relating s and CDE so we can use \[ABC = r^2CDE\]
anonymous
  • anonymous
oh, ok
anonymous
  • anonymous
I'm going to think about it for a bit lol
anonymous
  • anonymous
ok,thx
phi
  • phi
If you want a hint. I hope this is not too cryptic
1 Attachment
anonymous
  • anonymous
Can you clarify ? |dw:1327797362393:dw|
anonymous
  • anonymous
why is it 2/5?
phi
  • 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
anonymous
  • anonymous
oh yeah. Thank you.
anonymous
  • anonymous
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
anonymous
  • anonymous
Thank you, guys!!!

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