1. 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.

2. anonymous

|dw:1327792772445:dw|

3. 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.

4. anonymous

Yes

5. anonymous

What is the connection between those triangles?

6. 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.

7. 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?

8. anonymous

is it some sort of formula?

9. anonymous

$0.16s^2CFG = CFG \Longrightarrow s=\sqrt{\frac{1}{0.16}} = 2.5$ Also, $s=\frac{12}{FG} \Longrightarrow FG = 4.8$

10. anonymous

well similar lengths are scaled by a factor, and similar areas are scaled by the square of that factor

11. anonymous

ok so we have found s, maybe try to find r next

12. anonymous

oh, i see.

13. anonymous

so does it work the same for r too?

14. anonymous

|dw:1327795417846:dw|

15. anonymous

r is a little more tricky...

16. anonymous

because we don't know what percent is CDE of ABC

17. anonymous

actually no... $(0.4)^2 ABh_1 = FGh_2$ So it must be that $0.4h_1 = h_2$

18. anonymous

19. anonymous

Yeah, we need to try to figure out something relating s and CDE so we can use $ABC = r^2CDE$

20. anonymous

oh, ok

21. anonymous

I'm going to think about it for a bit lol

22. anonymous

ok,thx

23. phi

If you want a hint. I hope this is not too cryptic

24. anonymous

Can you clarify ? |dw:1327797362393:dw|

25. anonymous

why is it 2/5?

26. 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

27. anonymous

oh yeah. Thank you.

28. 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

29. anonymous

Thank you, guys!!!