Question

which statement could be used to fill in the missing step in the proof of the pythagorean theorem?

1. find the area of the outer square.
2. find the sum of the areas of the four triangles with legs a and b added to the square with side c.
3. ?

set the area of the outer square equal to the area of the four triangles and the inner square, then simplify.
set up a proportion between the areas of the four triangles and the area of the inner square, then simplify.
add the area of the inner square to the area of the outer square, then simplify.
subtract the area of the inner square from the area of the outer square, then simplify.

Explanation:

Step1: Analyze the proof of Pythagorean theorem

The proof of the Pythagorean theorem using the area - comparison method. The area of the outer square can be calculated in two ways. One way is by using the formula for the area of a square with side length \(a + b\), so \(A=(a + b)^2=a^{2}+2ab + b^{2}\). The other way is by summing the areas of the four right - angled triangles (each with area \(\frac{1}{2}ab\)) and the inner square with side length \(c\), so \(A = 4\times\frac{1}{2}ab+c^{2}=2ab + c^{2}\).

Step2: Compare the two expressions for the area

To prove \(a^{2}+b^{2}=c^{2}\), we need to set the two expressions for the area of the outer square (one as the area of the large square \((a + b)^2\) and the other as the sum of the areas of the four triangles and the inner square \(4\times\frac{1}{2}ab + c^{2}\)) equal to each other. That is, set the area of the outer square equal to the area of the four triangles and the inner square, then simplify.

Answer:

Set the area of the outer square equal to the area of the four triangles and the inner square, then simplify.