This picture shows a proof of the Pythagorean theorem because

Question

The image illustrates the Pythagorean theorem using two different arrangements of four congruent right triangles within squares. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is commonly expressed as a² + b² = c², where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the other two sides. Figure 4-A:
Four right triangles, each with sides labeled 'a', 'b', and 'c', are arranged within a larger square in such a way that their hypotenuses form an inner square. The area of the larger square is (a+b)², while the area of the inner square is c². The remaining area consists of the four triangles, each with an area of ½ab. The equation can be derived as (a+b)² = c² + 4(½ab), which simplifies to a² + 2ab + b² = c² + 2ab. By subtracting 2ab from both sides, the Pythagorean theorem, a² + b² = c², is obtained.
Figure 4-B:
The same four right triangles are rearranged within a square of the same size. This time, they are positioned to form two squares, one with side 'a' and the other with side 'b'. The total area of these two squares is a² + b². Since the total area of the larger square is the same in both figures and the area of the four triangles is also the same, the remaining area in Figure 4-B must be equal to the area of the inner square in Figure 4-A. Therefore, a² + b² = c².

(1 point)
Responses

The four yellow triangles are congruent in both large squares and the large squares are congruent, so c2=a2+b2
The four yellow triangles are congruent in both large squares and the large squares are congruent, so c squared is equal to A squared plus b squared

a < b < c, so the Pythagorean theorem holds true.
a < b < c, so the Pythagorean theorem holds true.

The yellow triangles fill up the large squares just enough to have a small square in each figure.
The yellow triangles fill up the large squares just enough to have a small square in each figure.

a = b = c, so the Pythagorean theorem holds true.

Answer 1

The correct response regarding the proof of the Pythagorean theorem illustrated in the described figures is:

The four yellow triangles are congruent in both large squares and the large squares are congruent, so c² = a² + b².

This response accurately summarizes the reasoning behind the proof. The arrangement of the congruent triangles within both configurations allows for the visualization that the area of the larger square equals the combined areas of the squares formed by the legs of the right triangles (a and b) plus the area of the triangles, reinforcing the relationship stated in the Pythagorean theorem, \( a² + b² = c² \).