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DarlingDare Group Title

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

  • one year ago
  • one year ago

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  1. DarlingDare Group Title
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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

    • one year ago
  2. DarlingDare Group Title
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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.

    • one year ago
  3. DarlingDare Group Title
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    Please help

    • one year ago
  4. DarlingDare Group Title
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    May you help me? Please?????

    • one year ago
  5. myininaya Group Title
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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

    • one year ago
  6. DarlingDare Group Title
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    Ok

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

    • one year ago
  8. myininaya Group Title
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    Ok those are the areas of the big shape that contains all the small shapes for each

    • one year ago
  9. myininaya Group Title
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    Now find the areas of the smaller shapes inside the big shapes

    • one year ago
  10. DarlingDare Group Title
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    Ok.

    • one year ago
  11. DarlingDare Group Title
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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.

    • one year ago
  12. myininaya Group Title
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    and there are four of those triangles right?

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

    • one year ago
  14. myininaya Group Title
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    \[bigger area=the \small \square+the triangles \]

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

    • one year ago
  16. myininaya Group Title
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    For gina's

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

    • one year ago
  18. DarlingDare Group Title
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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.

    • one year ago
  19. myininaya Group Title
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    Ok yep you are right gina messed up there :)

    • one year ago
  20. myininaya Group Title
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    Ok now looking at sean's

    • one year ago
  21. myininaya Group Title
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    you said the area of big square is c^2 right?

    • one year ago
  22. myininaya Group Title
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    now lets find the areas of the smaller shapes that are contained in this big square

    • one year ago
  23. DarlingDare Group Title
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    Of the bigger square, yes.

    • one year ago
  24. DarlingDare Group Title
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    Ok

    • one year ago
  25. myininaya Group Title
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    the area of the small square is?

    • one year ago
  26. DarlingDare Group Title
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    a-b^2

    • one year ago
  27. myininaya Group Title
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    (a-b)^2 is right

    • one year ago
  28. DarlingDare Group Title
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    (a-b)(a-b) ?

    • one year ago
  29. myininaya Group Title
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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

    • one year ago
  30. myininaya Group Title
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    right (a-b)^2=(a-b)(a-b) :)

    • one year ago
  31. DarlingDare Group Title
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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.

    • one year ago
  32. myininaya Group Title
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    1/2 *ab right?

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

    • one year ago
  34. myininaya Group Title
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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

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

    • one year ago
  36. DarlingDare Group Title
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    c^2 = 2ab + (a-b)(a-b) c^2 =2ab+ a^2 -ab-ab+b

    • one year ago
  37. myininaya Group Title
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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?

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

    • one year ago
  39. DarlingDare Group Title
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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.

    • one year ago
  40. myininaya Group Title
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    Hmmm then I must be looking at something different

    • one year ago
  41. myininaya Group Title
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    Are you sure? look again.

    • one year ago
  42. DarlingDare Group Title
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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"

    • one year ago
  43. myininaya Group Title
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    The ordering he has is a little different

    • one year ago
  44. DarlingDare Group Title
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    Ohhh..... whoops.

    • one year ago
  45. DarlingDare Group Title
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    Ok, then he's right. I just didn't see it correctly.... Thanks!!!

    • one year ago
  46. myininaya Group Title
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    :) So how do you feel about this question? Do you understand it better?

    • one year ago
  47. DarlingDare Group Title
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    Yes. Thank you so very much!!!!!

    • one year ago
  48. DarlingDare Group Title
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    :)

    • one year ago
  49. myininaya Group Title
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    Np. Have a great day.

    • one year ago
  50. DarlingDare Group Title
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    You too!

    • one year ago
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