The following two column proof with missing justifications proves the Pythagorean

The following two column proof with missing justifications proves the Pythagorean? Draw an altitude from point C to AB^- Let BC^- = a CA^- = b AB^- = c CD^- = h DB^- = y AD^- = x y – x = c c/a = a/y; c/b = b/x a^2 = cy; b^2 = cx a^2 + b^2 = cy + cx a^2 + b^2 = c(y + x) a^2 + b^2 = c(c) a^2 + b^2 = c^2 Which is not a justification for the proof? Pieces of Right Triangles Similarity Theorem Side Side Side Similarity Theorem Substitution Addition Property of Equality

The Correct Answer and Explanation is :

The missing justification in the proof provided is “Side Side Side Similarity Theorem”.

Explanation:

In the given two-column proof, the goal is to prove the Pythagorean theorem using the concept of right triangles and the relationships between the sides. Let’s break down the justification process:

1. Draw an altitude from point C to AB: This step creates two right triangles, triangle ABC and triangle ACD, which will help establish relationships between the sides.
2. BC = a, CA = b, AB = c, CD = h, DB = y, AD = x: These are the given lengths of the sides and the parts of the triangles.
3. y – x = c: This equation represents the relationship between the segments formed by the altitude.
4. c/a = a/y and c/b = b/x: These two ratios are based on the similarity of the triangles. Since the triangles are right triangles and share angle C, they are similar by the AA (Angle-Angle) similarity criterion, not by Side Side Side (SSS) similarity. This is a crucial distinction: similarity of triangles involves having the same angle measures, and proportionality of corresponding sides, not just matching side lengths.
5. a^2 = cy, b^2 = cx: These equations follow from applying the geometric mean (altitude) theorem, not from SSS similarity.
6. a^2 + b^2 = cy + cx: Here, addition of the above equations is justified by the Addition Property of Equality.
7. a^2 + b^2 = c(y + x): Using substitution, we know that (y + x = c) from earlier, so this equation is valid.
8. a^2 + b^2 = c^2: Finally, the proof concludes by simplifying the equation, thus proving the Pythagorean theorem.

Why “Side Side Side Similarity Theorem” is Incorrect:

• The Side Side Side (SSS) Similarity Theorem is used when two triangles have corresponding sides in the same ratio. However, in this proof, we are using the Angle-Angle (AA) similarity criterion to establish that the triangles are similar. Therefore, the SSS similarity does not apply here.

The other justifications, such as Substitution, Addition Property of Equality, and the Pieces of Right Triangles Similarity Theorem, are correct and appropriately applied in the proof.

Image:

Here is an illustration of the two right triangles (ABC and ACD) with the altitude from C to AB to visualize the relationships between the sides.

Here is a geometric diagram showing the right triangle ABC with the altitude CD drawn from point C to side AB. The sides are labeled as follows: BC = a, CA = b, AB = c, CD = h, DB = y, and AD = x. The diagram also demonstrates that y + x = c, with the triangles being similar by AA similarity, not SSS similarity, as well as showing the key relationship a^2 + b^2 = c^2.