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anonymous

  • one year ago

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.

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  1. anonymous
    • one year ago
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  2. anonymous
    • one year ago
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    I just need help on how to solve the proof

  3. anonymous
    • one year ago
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    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.

  4. anonymous
    • one year ago
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    I know number one is given

  5. anonymous
    • one year ago
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    For number 2 would it be AC/BC=BC/AC?

  6. anonymous
    • one year ago
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    The first reason would be "Given." We are given this information in the problem. The second statement would be: $$\frac{AC}{AB}=\frac{AD}{AC}$$

  7. anonymous
    • one year ago
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    ah ok

  8. anonymous
    • one year ago
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    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

  9. anonymous
    • one year ago
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    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.

  10. anonymous
    • one year ago
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    so then the third would be b/c=e/b

  11. anonymous
    • one year ago
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    Yes. Now cross-multiply. $$\frac{b}{c}=\frac{e}{b}$$ So b^2 = ce

  12. anonymous
    • one year ago
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    Cross multiply would be the reason for number four. Five is complete. Six is going to be similar to two.

  13. anonymous
    • one year ago
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    would it be BC/BA=BD/BC

  14. anonymous
    • one year ago
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    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.

  15. anonymous
    • one year ago
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    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

  16. anonymous
    • one year ago
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    Now, if we add the equation in statement four to that in statement eight, we get: $$a^2+b^2=cd+ce$$

  17. anonymous
    • one year ago
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    What would be for number 10 then?

  18. anonymous
    • one year ago
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    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

  19. anonymous
    • one year ago
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    The factoring would be number 10.

  20. anonymous
    • one year ago
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    which is the Pythagorean theorem THANKS!

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