Answer :
To determine why the triangles are similar for the proof of the Pythagorean theorem using similarity, let's explore the given options:
1. The geometric mean (altitude) theorem: This theorem states that in a right triangle, the altitude drawn to the hypotenuse creates two smaller triangles that are similar to each other and to the original triangle. This theorem deals with the altitude specifically.
2. The geometric mean (leg) theorem: This theorem says that in a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg. This is also a similarity theorem but focuses on the legs.
3. The right triangle altitude theorem: This theorem states that if you draw an altitude from the right angle of a right triangle to the hypotenuse, the two resulting triangles are similar to the original triangle and to each other. This is a more general form of the geometric mean theorems, encompassing the entire structure of the right triangle's similarity properties.
4. The SSS theorem: The Side-Side-Side theorem states that if all three sides of one triangle are proportional to all three sides of another triangle, then the triangles are similar. This theorem, although useful in proving similarity, does not specifically address the properties unique to right triangles and the altitude that proves the Pythagorean theorem.
Given these facts, the correct answer that allows us to state the required proportions and proves the similarity of the triangles in the context of the Pythagorean theorem is:
The right triangle altitude theorem
1. The geometric mean (altitude) theorem: This theorem states that in a right triangle, the altitude drawn to the hypotenuse creates two smaller triangles that are similar to each other and to the original triangle. This theorem deals with the altitude specifically.
2. The geometric mean (leg) theorem: This theorem says that in a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg. This is also a similarity theorem but focuses on the legs.
3. The right triangle altitude theorem: This theorem states that if you draw an altitude from the right angle of a right triangle to the hypotenuse, the two resulting triangles are similar to the original triangle and to each other. This is a more general form of the geometric mean theorems, encompassing the entire structure of the right triangle's similarity properties.
4. The SSS theorem: The Side-Side-Side theorem states that if all three sides of one triangle are proportional to all three sides of another triangle, then the triangles are similar. This theorem, although useful in proving similarity, does not specifically address the properties unique to right triangles and the altitude that proves the Pythagorean theorem.
Given these facts, the correct answer that allows us to state the required proportions and proves the similarity of the triangles in the context of the Pythagorean theorem is:
The right triangle altitude theorem
