## 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

1. DarlingDare Group Title

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 Group Title
3. DarlingDare Group Title
4. DarlingDare Group Title

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.

5. DarlingDare Group Title

6. DarlingDare Group Title

7. myininaya Group Title

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

8. DarlingDare Group Title

Ok

9. DarlingDare Group Title

So Sean's figure would be c^2, and Gina's figure would be (a+b)^2.

10. myininaya Group Title

Ok those are the areas of the big shape that contains all the small shapes for each

11. myininaya Group Title

Now find the areas of the smaller shapes inside the big shapes

12. DarlingDare Group Title

Ok.

13. DarlingDare Group Title

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

14. myininaya Group Title

and there are four of those triangles right?

15. DarlingDare Group Title

And there are 4 triangles in Gina's figure so it would be 2 ab.

16. myininaya Group Title

$bigger area=the \small \square+the triangles$

17. myininaya Group Title

Yep yep :) So we have $(a+b)^2=c^2+2ab \text{ right?}$

18. myininaya Group Title

For gina's

19. myininaya Group Title

Now expand (a+b)^2 by writing (a+b)(a+b) and then distributing (or multiplying)

20. DarlingDare Group Title

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.

21. myininaya Group Title

Ok yep you are right gina messed up there :)

22. myininaya Group Title

Ok now looking at sean's

23. myininaya Group Title

you said the area of big square is c^2 right?

24. myininaya Group Title

now lets find the areas of the smaller shapes that are contained in this big square

25. DarlingDare Group Title

Of the bigger square, yes.

26. DarlingDare Group Title

Ok

27. myininaya Group Title

the area of the small square is?

28. DarlingDare Group Title

a-b^2

29. myininaya Group Title

(a-b)^2 is right

30. DarlingDare Group Title

(a-b)(a-b) ?

31. myininaya Group Title

Now tell me the area of on those triangle and then we will multiply it by 4 since there are 4 triangles

32. myininaya Group Title

right (a-b)^2=(a-b)(a-b) :)

33. DarlingDare Group Title

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.

34. myininaya Group Title

1/2 *ab right?

35. DarlingDare Group Title

Yes... I forgot the 1/2 part... whoops. Thanks for helping me remember that! :)

36. myininaya Group Title

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

37. myininaya Group Title

$c^2=4 \cdot \frac{1}{2} \cdot ab+(a-b)^2$

38. DarlingDare Group Title

c^2 = 2ab + (a-b)(a-b) c^2 =2ab+ a^2 -ab-ab+b

39. myininaya Group Title

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?

40. myininaya Group Title

mean because -ab-ab=-2ab and so you have c^2=2ab+a^2-2ab+b^2

41. DarlingDare Group Title

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

42. myininaya Group Title

Hmmm then I must be looking at something different

43. myininaya Group Title

Are you sure? look again.

44. DarlingDare Group Title

It says, "Step 3: Area of EFGH = c2 = Area of 4 triangles + area of ABCD = 2ab + a2 + b2 – 2ab Hence c2 = a2 + b2"

45. myininaya Group Title

The ordering he has is a little different

46. DarlingDare Group Title

Ohhh..... whoops.

47. DarlingDare Group Title

Ok, then he's right. I just didn't see it correctly.... Thanks!!!

48. myininaya Group Title

49. DarlingDare Group Title

Yes. Thank you so very much!!!!!

50. DarlingDare Group Title

:)

51. myininaya Group Title

Np. Have a great day.

52. DarlingDare Group Title

You too!