Question

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

Answer 1

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

Answer 2

http://learn.flvs.net/webdav/assessment_images/educator_geometry_v14/pool_Geom_3641_1000_Subtest_02_05/image0054e8c5217.jpg

Answer 3

http://learn.flvs.net/webdav/assessment_images/educator_geometry_v14/pool_Geom_3641_1000_Subtest_02_05/image0054e8c5217.jpg

Answer 4

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.

Answer 5

Please help

Answer 6

May you help me?
Please?????

Answer 7

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

Answer 8

Ok

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

Ok.

Answer 13

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

Answer 14

and there are four of those triangles right?

Answer 15

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

Answer 16

\[bigger area=the \small \square+the triangles \]

Answer 17

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

Answer 18

For gina's

Answer 19

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

Answer 20

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.

Answer 21

Ok yep you are right gina messed up there :)

Answer 22

Ok now looking at sean's

Answer 23

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

Answer 24

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

Answer 25

Of the bigger square, yes.

Answer 26

Ok

Answer 27

the area of the small square is?

Answer 28

a-b^2

Answer 29

(a-b)^2 is right

Answer 30

(a-b)(a-b) ?

Answer 31

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

Answer 32

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

Answer 33

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.

Answer 34

1/2 *ab right?

Answer 35

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

Answer 36

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

Answer 37

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

Answer 38

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

Answer 39

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?

Answer 40

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

Answer 41

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

Answer 42

Hmmm then I must be looking at something different

Answer 43

Are you sure? look again.

Answer 44

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

Answer 45

The ordering he has is a little different

Answer 46

Ohhh..... whoops.

Answer 47

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

Answer 48

:)
So how do you feel about this question?
Do you understand it better?

Answer 49

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

Answer 50

:)

Answer 51

Np. Have a great day.

Answer 52

You too!