To determine which principle allows us to state that the triangles are similar and consequently write the true proportions, let's first understand the key concepts and then identify the correct theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse [tex]\( c \)[/tex] is equal to the sum of the squares of the other two sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
When proving the Pythagorean theorem using similarity, we often draw an altitude from the right angle vertex to the hypotenuse of the original right triangle. This construction leads to the formation of two smaller right triangles within the original triangle. The two smaller triangles and the original triangle all share the same angle measures, which implies that they are similar by the Angle-Angle (AA) criterion for similarity of triangles.
Theorems relevant to such constructions include:
1. The geometric mean (altitude) theorem: This theorem states that the altitude drawn to the hypotenuse of a right triangle creates two segments on the hypotenuse. The altitude is the geometric mean between the lengths of these two segments. Additionally, it implies that each of the two smaller right triangles is similar to the original right triangle and to each other.
2. The geometric mean (leg) theorem: This theorem states that each leg of a right triangle is the geometric mean between the hypotenuse and the projection of the leg on the hypotenuse.
3. The right triangle altitude theorem: This is another way to refer to the geometric mean (altitude) theorem.
4. The SSS theorem: This stands for Side-Side-Side similarity, which is not typically used in the context of proving the Pythagorean theorem with altitude constructions and similarity.
To use similarity to write proportions such as [tex]\(\frac{c}{a} = \frac{G}{f}\)[/tex] and [tex]\(\frac{c}{b} = \frac{b}{c}\)[/tex], we rely on the geometric mean (altitude) theorem, as it deals directly with the relationships between the segments of the hypotenuse and the created similar right triangles.
Therefore, we can conclude that the theorem which allows us to state that the triangles are similar in order to write the true proportions is:
The geometric mean (altitude) theorem
