anonymous
  • anonymous
Rex, Paulo, and Ben are standing on the shore watching for dolphins. Paulo sees one surface directly in front of him about a hundred feet away. Use the spaces provided below to prove that the square of the distance between Rex and Ben is the same as the sum of the squares of the distances between Rex and the dolphin, and Ben and the dolphin.
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
anonymous
  • anonymous
I just need help on how to solve the proof
anonymous
  • anonymous
In general, there is a theorem that would be useful here and perhaps allow the skipping of a couple steps (geometric mean leg theorem if I'm not mistaken). But the proof of that theorem relies on similar triangles, which is what is given in this problem anyway.

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anonymous
  • anonymous
I know number one is given
anonymous
  • anonymous
For number 2 would it be AC/BC=BC/AC?
anonymous
  • anonymous
The first reason would be "Given." We are given this information in the problem. The second statement would be: $$\frac{AC}{AB}=\frac{AD}{AC}$$
anonymous
  • anonymous
ah ok
anonymous
  • anonymous
The reason for this is that AC is the hypotenuse of the small triangle. So it corresponds to AB, which is the hypotenuse of the biggest triangle
anonymous
  • anonymous
Now, AC is also the smallest side of the biggest triangle, so we want to find the smallest side of the small triangle (ADC). This is AD.
anonymous
  • anonymous
so then the third would be b/c=e/b
anonymous
  • anonymous
Yes. Now cross-multiply. $$\frac{b}{c}=\frac{e}{b}$$ So b^2 = ce
anonymous
  • anonymous
Cross multiply would be the reason for number four. Five is complete. Six is going to be similar to two.
anonymous
  • anonymous
would it be BC/BA=BD/BC
anonymous
  • anonymous
For six, it may help to look at seven and work backwards. Yes, you're right. The ratio of the longer leg to the hypotenuse in the biggest triangle equals the ratio of the longer leg to hypotenuse in the second-biggest triangle.
anonymous
  • anonymous
So the reason for six would be the same as that for two. The reason for seven will be the same as that for three. In eight, cross-multiply to get that a^2=cd
anonymous
  • anonymous
Now, if we add the equation in statement four to that in statement eight, we get: $$a^2+b^2=cd+ce$$
anonymous
  • anonymous
What would be for number 10 then?
anonymous
  • anonymous
Now factor out c on the right side. a^2+b^2 = c(d+e) But d + e is c. So a^2 + b^2 = c*c = c^2
anonymous
  • anonymous
The factoring would be number 10.
anonymous
  • anonymous
which is the Pythagorean theorem THANKS!

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