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I feel very stupid, but I have to answer this question but I cannot seem to solve it! :( I have to find the length of DF. I already figured out that because angle C = angle A1 (left part of the angle) Thales applies so AD must be 12 as well. But I don't know how to proceed. Anyone has a clue? Thanks in advance!

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Sanne
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  • This might help : http://en.wikipedia.org/wiki/Law_of_cosines – pitchounet Jan 09 '14 at 13:07
  • Thank you! I also thought of that, but I have to explain this to my little brother (and as I study physics I should know this). But he has not discussed cosines yet! In class he only discussed Thales and similarity, so that is basically the toolbox I can use.. – Sanne Jan 09 '14 at 13:12
  • Is the angle at A 90°? I wonder that you say "Thales applies" ... – Bernd Jan 09 '14 at 13:13
  • Yes, because AD is 12 (because C = A1), Thales applies, implying A is 90 degrees – Sanne Jan 09 '14 at 13:14
  • I thought of this? http://en.wikipedia.org/wiki/Thales%27_theorem – Sanne Jan 09 '14 at 13:15
  • Sorry, but could you please explain again what is given and how you deduct AD = 12? Is CD = DB = 12 given? – Bernd Jan 09 '14 at 13:15
  • Okay, sorry if I am not so clear. In the question it is just given that BC=24, and that CD=BD and AE=BE and that the angle C= left part of angle A. – Sanne Jan 09 '14 at 13:17
  • Then I deduced (maybe wrongly?) that AD is 12 as well, and the total angle A should be 90 degrees. – Sanne Jan 09 '14 at 13:17
  • That's fine. See Answer from Traklon – Bernd Jan 09 '14 at 13:25

2 Answers2

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If you know $AD = 12$ already, then $DF = 12/3$. This is because $F$ is the centroid of $\triangle ABC$, and the centroid of a triangle divides each median into $1:2$.

peterwhy
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F is the intersection of all three medians of the triangle. You can know the length of AD (which is a median) with http://en.wikipedia.org/wiki/Apollonius%27_theorem if you have the length of the 3 sides (in ABC), or http://en.wikipedia.org/wiki/Law_of_cosines if you have 2 angles and 1 length (in ACD), which you seem to have. After that, it is a result that the length of DF is equal to the third of the length of AD.

EDIT : I just saw that ABC is rectangle. That simplifies the problem of knowing the length of AD if you know that if ABC is rectangle in A, BC is the diameter of a circle passing by A, B and C. What relation can you deduce between AD and BD ?

Traklon
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