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DarlingDare

  • 2 years ago

Gina and Sean drew the following figures to prove the Pythagorean Theorem c2 = a2 + b2. Both figures are made of two squares and four identical right triangles, as shown below. Gina and Sean wrote the following proofs. Gina’s Proof: Step 1: Area of PQRS = (a + b) 2 = a2 + b2 + 4ab Step 2: Area of triangle PKN =; hence the area of 4 triangles = 4ab Step 3: Area of KLMN = c2 = area of PQRS – area of 4 triangles = a2 + b2 + 4ab – 4ab Hence, c2 = a2 + b2

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  1. DarlingDare
    • 2 years ago
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    Sean’s Proof: Step 1: Area of triangle EAF =; hence the area of 4 triangles = Step 2: Area of square ABCD = (a – b) 2 = a2 + b2 – 2ab Step 3: Area of EFGH = c2 = Area of 4 triangles + area of ABCD = 2ab + a2 + b2 – 2ab Hence c2 = a2 + b2

  2. DarlingDare
    • 2 years ago
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    Which statement gives the correct conclusion about the proofs given by Gina and Sean? Sean’s proof is correct and Gina’s proof is incorrect. Both the proofs are correct. Gina’s proof is correct and Sean’s proof is incorrect. Both the proofs are incorrect.

  3. DarlingDare
    • 2 years ago
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    Please help

  4. DarlingDare
    • 2 years ago
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    May you help me? Please?????

  5. myininaya
    • 2 years ago
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    Look at one square at a time. Find the area of the bigger shape using the formula for area of square. Then find the area of the smaller shapes that are contained in the bigger shape. And then see if you get the Pythagorean thm c^2=a^2+b^2

  6. DarlingDare
    • 2 years ago
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    Ok

  7. DarlingDare
    • 2 years ago
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    So Sean's figure would be c^2, and Gina's figure would be (a+b)^2.

  8. myininaya
    • 2 years ago
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    Ok those are the areas of the big shape that contains all the small shapes for each

  9. myininaya
    • 2 years ago
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    Now find the areas of the smaller shapes inside the big shapes

  10. DarlingDare
    • 2 years ago
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    Ok.

  11. DarlingDare
    • 2 years ago
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    The square in Gina's figue would be c^2, and the triangles in Gina's figure will be 1/2 ab.

  12. myininaya
    • 2 years ago
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    and there are four of those triangles right?

  13. DarlingDare
    • 2 years ago
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    And there are 4 triangles in Gina's figure so it would be 2 ab.

  14. myininaya
    • 2 years ago
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    \[bigger area=the \small \square+the triangles \]

  15. myininaya
    • 2 years ago
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    Yep yep :) So we have \[(a+b)^2=c^2+2ab \text{ right?}\]

  16. myininaya
    • 2 years ago
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    For gina's

  17. myininaya
    • 2 years ago
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    Now expand (a+b)^2 by writing (a+b)(a+b) and then distributing (or multiplying)

  18. DarlingDare
    • 2 years ago
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    Yes. But, looking at the diagram, we know she isn't right, because she put 4 ab, instead of 2ab. So it's either Sean's right, or neither of them are right.

  19. myininaya
    • 2 years ago
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    Ok yep you are right gina messed up there :)

  20. myininaya
    • 2 years ago
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    Ok now looking at sean's

  21. myininaya
    • 2 years ago
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    you said the area of big square is c^2 right?

  22. myininaya
    • 2 years ago
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    now lets find the areas of the smaller shapes that are contained in this big square

  23. DarlingDare
    • 2 years ago
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    Of the bigger square, yes.

  24. DarlingDare
    • 2 years ago
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    Ok

  25. myininaya
    • 2 years ago
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    the area of the small square is?

  26. DarlingDare
    • 2 years ago
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    a-b^2

  27. myininaya
    • 2 years ago
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    (a-b)^2 is right

  28. DarlingDare
    • 2 years ago
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    (a-b)(a-b) ?

  29. myininaya
    • 2 years ago
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    Now tell me the area of on those triangle and then we will multiply it by 4 since there are 4 triangles

  30. myininaya
    • 2 years ago
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    right (a-b)^2=(a-b)(a-b) :)

  31. DarlingDare
    • 2 years ago
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    ab, since on the hypotenuse, there are two different line segments on the same line which is a-b+b, times b. or otherwise, a times b.

  32. myininaya
    • 2 years ago
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    1/2 *ab right?

  33. DarlingDare
    • 2 years ago
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    Yes... I forgot the 1/2 part... whoops. Thanks for helping me remember that! :)

  34. myininaya
    • 2 years ago
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    And now there are four of those triangles so We have : for sean: c^2=4*1/2*ab+(a-b)^2 bigsquare=thetriangles+smallsquare

  35. myininaya
    • 2 years ago
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    \[c^2=4 \cdot \frac{1}{2} \cdot ab+(a-b)^2\]

  36. DarlingDare
    • 2 years ago
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    c^2 = 2ab + (a-b)(a-b) c^2 =2ab+ a^2 -ab-ab+b

  37. myininaya
    • 2 years ago
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    well one little correction to what you have: c^2 = 2ab + (a-b)(a-b) c^2 =2ab+ a^2 -ab-ab+b^2 Step 3: Area of EFGH = c2 = Area of 4 triangles + area of ABCD = 2ab + a2 + b2 – 2ab And that is what he has for step 3 correct?

  38. myininaya
    • 2 years ago
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    mean because -ab-ab=-2ab and so you have c^2=2ab+a^2-2ab+b^2

  39. DarlingDare
    • 2 years ago
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    no... he's not correct, because he didn't have the a^2, or the negative sign in front of 2ab.

  40. myininaya
    • 2 years ago
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    Hmmm then I must be looking at something different

  41. myininaya
    • 2 years ago
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    Are you sure? look again.

  42. DarlingDare
    • 2 years ago
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    It says, "Step 3: Area of EFGH = c2 = Area of 4 triangles + area of ABCD = 2ab + a2 + b2 – 2ab Hence c2 = a2 + b2"

  43. myininaya
    • 2 years ago
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    The ordering he has is a little different

  44. DarlingDare
    • 2 years ago
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    Ohhh..... whoops.

  45. DarlingDare
    • 2 years ago
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    Ok, then he's right. I just didn't see it correctly.... Thanks!!!

  46. myininaya
    • 2 years ago
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    :) So how do you feel about this question? Do you understand it better?

  47. DarlingDare
    • 2 years ago
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    Yes. Thank you so very much!!!!!

  48. DarlingDare
    • 2 years ago
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    :)

  49. myininaya
    • 2 years ago
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    Np. Have a great day.

  50. DarlingDare
    • 2 years ago
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    You too!

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