To justify the similarity of the triangles in this proof of the Pythagorean theorem, we can utilize a specific geometric property.
In a right triangle, if you drop an altitude from the right angle to the hypotenuse, you create two smaller right triangles within the original triangle. These two smaller triangles are not just right triangles themselves, but they are also similar to each other and to the original triangle. This principle is known as the Right Triangle Altitude Theorem.
Here's a detailed explanation of the steps:
1. Right Triangle and Altitude: Consider a right triangle [tex]\( \triangle ABC \)[/tex] with [tex]\( \angle C \)[/tex] being the right angle. Drop an altitude [tex]\( CD \)[/tex] from the right angle [tex]\( C \)[/tex] to the hypotenuse [tex]\( AB \)[/tex]. This creates two smaller triangles, [tex]\( \triangle ACD \)[/tex] and [tex]\( \triangle CBD \)[/tex].
2. Similarity of Triangles: According to the Right Triangle Altitude Theorem, [tex]\( \triangle ACD \sim \triangle ABC \)[/tex] and [tex]\( \triangle CBD \sim \triangle ABC \)[/tex]. Since [tex]\( \triangle ACD \)[/tex] and [tex]\( \triangle CBD \)[/tex] are both similar to [tex]\( \triangle ABC \)[/tex], they are also similar to each other ([tex]\( \triangle ACD \sim \triangle CBD \)[/tex]).
3. Proportional Relationships: The similarity of these triangles allows us to set up proportional relationships between their corresponding sides. For the triangles' sides, these proportions hold:
[tex]\[
\frac{c}{a} = \frac{a}{f} \quad \text{and} \quad \frac{c}{b} = \frac{b}{e}
\][/tex]
where:
- [tex]\( c \)[/tex] is the hypotenuse of the original right triangle [tex]\( ABC \)[/tex],
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the original right triangle,
- [tex]\( e \)[/tex] and [tex]\( f \)[/tex] are segments into which the altitude divides the hypotenuse [tex]\( AB \)[/tex].
Therefore, these proportions are valid and derived from the fact that the triangles [tex]\( \triangle ACD \)[/tex], [tex]\( \triangle CBD \)[/tex], and [tex]\( \triangle ABC \)[/tex] are all similar. The similarity is specifically established by the Right Triangle Altitude Theorem. This theorem is fundamental in writing the true proportions required in the proof of the Pythagorean theorem.
Thus, the correct justification for stating that the triangles are similar to write the proportions [tex]\(\frac{c}{a}=\frac{a}{f}\)[/tex] and [tex]\(\frac{c}{b}=\frac{b}{e}\)[/tex] is provided by the Right Triangle Altitude Theorem.
