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
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.