Question

Use the image to answer the question.



Betty is writing a proof of the Pythagorean Theorem using the diagram shown. In the diagram, there is a large square. A smaller square is drawn inside the large square, dividing the large square into a smaller square and 4 small triangles.

Step 1: First, I know that the area of a triangle is equal to bh2

ℎ
2
. The area of each small triangle would be ab2


2
. Since there are 4 triangles in the entire square, the area of all 4 triangles together would be equal to 2ab
2


. The area of the small square in the middle can be represented by c2

2
since the sides each have a length of c

. Therefore, the area of the entire large square can be written as 2ab+c2
2


+

2
.

Step 2: Another way to think about the area of the entire large square is by looking at its side lengths. The length of each side can be written as a+b

+

, so the area would be (a+b)×(a+b)
(

+

)
×
(

+

)
.

Step 3: Since we've expressed the area of the large square in two different ways, now set those two areas equal to each other and then simplify using algebra.

(a+b)×(a+b)=2ab+c2

Answer 1

So what is the answer?

Answer 2

Step 4: By expanding the expression on the left side, we get a^2 + 2ab + b^2. Setting this equal to 2ab + c^2, we have a^2 + 2ab + b^2 = 2ab + c^2.

Step 5: By rearranging the terms, we have a^2 + b^2 = c^2, which is the Pythagorean Theorem. Therefore, the proof of the Pythagorean Theorem is complete using the given diagram.

Answer 3

The answer is that Betty was able to prove the Pythagorean Theorem using the given diagram, and the final equation derived from the proof is a^2 + b^2 = c^2, which is the Pythagorean Theorem.