Question

Given a right triangle ABC, where BC is the hypotenuse and a, b, and c are the lengths of the sides. Perpendicular CD is drawn from vertex C to hypotenuse AB, forming right triangles BDC and CDA, with CD = h, BD = y, and DA = x.

The proof aims to establish the Pythagorean theorem a²+b²=c² using similar triangles. The steps of the proof involve constructing an altitude, labeling segments, applying the segment addition postulate, and using the cross product property, addition property of equality, and substitution.

Which justification is missing in the proof?

a) Substitution
b) Addition Property of Equality
c) Pieces of Right Triangles
d) Similarity Theorem
e) Transitive Property of Equality

Answer 1

The Similarity Theorem is missing in the proof that uses similar triangles to establish the Pythagorean theorem, as it is vital for justifying the equating of side ratios in similar triangles.

Step-by-step explanation:

The justification missing in the proof of the Pythagorean theorem using similar triangles is the Similarity Theorem. When constructing a proof to establish that a² + b² = c² for a right triangle with hypotenuse c and legs a and b, the Similarity Theorem is crucial. It is used to show that the smaller triangles formed by drawing altitude CD are similar to the original right triangle ABC, which allows the properties of similar triangles to be applied. Without establishing the similarity of these triangles, the cross products of the sides cannot be properly equated, and the rest of the proof would not follow logically.