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Assignment: Justifying Steps in the Proof of the Pythagorean Theorem

In a proof of the Pythagorean theorem using similar triangles, what allows you to state that the triangles are similar in order to write the true proportions [tex]$\frac{c}{a} = \frac{a}{f}$[/tex] and [tex]$\frac{c}{b} = \frac{b}{e}$[/tex]?

A. The geometric mean (altitude) theorem
B. The geometric mean (leg) theorem
C. The right triangle altitude theorem
D. The SSS theorem

Sagot :

GinnyAnswer
To determine which theorem justifies the steps in the proof of the Pythagorean theorem involving similar triangles, we need to recall the definition and the properties of the relevant theorems:

1. Geometric Mean (Altitude) Theorem: This theorem states that in a right triangle, the altitude drawn to the hypotenuse creates two right triangles that are similar to each other and to the original triangle. This theorem helps us to write proportions involving the altitude.

2. Geometric Mean (Leg) Theorem: This theorem tells us that in a right triangle, the length of a leg is the geometric mean of the hypotenuse and the projection of that leg on the hypotenuse.

3. Right Triangle Altitude Theorem: This theorem asserts that the altitude drawn to the hypotenuse of a right triangle creates two smaller right triangles that are each similar to the original right triangle and to each other.

4. SSS (Side-Side-Side) Theorem: This is a congruence criterion that states if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Given the context of using similar triangles to derive the proportions in the proof of the Pythagorean theorem, let's analyze which theorem is the most appropriate.

In the context of the Pythagorean theorem, the important aspect is how drawing an altitude from the right angle to the hypotenuse of the original right triangle creates two smaller right triangles. These smaller right triangles are then similar to the original triangle and to each other. This similarity allows us to set up proportions between the sides of these triangles.

Therefore, the key element is the creation of similar triangles through the altitude, not the congruence of sides.

Thus, the theorem that justifies the similarity of the triangles in this context, allowing us to write true proportions like [tex]\(\frac{c}{a} = \frac{a}{f}\)[/tex] or [tex]\(\frac{c}{b} = \frac{b}{g}\)[/tex], is the right triangle altitude theorem.

Hence, the correct theorem that allows you to state that the triangles are similar in order to write the true proportions for the proof of the Pythagorean theorem is:

The right triangle altitude theorem.

The answer is 3.
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