Question

Given: δabc is a right triangle. prove: a2 b2 = c2 right triangle bca with sides of length a, b, and c. perpendicular cd forms right triangles bdc and cda. cd measures h units, bd measures y units, da measures x units. the following two-column proof proves the pythagorean theorem using similar triangles. statement justification draw an altitude from point c to line segment ab by construction let segment bc = a segment ca = b segment ab = c segment cd = h segment db = y segment ad = x by labeling y x = c segment addition postulate c over a equals a over y and c over b equals b over x ? a2 = cy; b2 = cx cross product property a2 b2 = cy b2 addition property of equality a2 b2 = cy cx substitution a2 b2 = c(y x) distributive property of equality a2 b2 = c(c) substitution a2 b2 = c2 multiplication which of the following is the missing justification in the proof? substitution addition property of equality pieces of right triangles similarity theorem transitive property of equality

Answer 1

The missing justification in the proof of the Pythagorean theorem using similar triangles is the similarity theorem, which confirms the proportional relationship between sides of similar triangles and leads to the conclusion a² + b² = c².

Step-by-step explanation:

The missing justification in the two-column proof that uses similar triangles to prove the Pythagorean theorem is the similarity theorem. The steps involving c/a = a/y and c/b = b/x rely on the fact that the smaller triangles ΔBDC and ΔCDA are similar to the original right triangle ΔABC. This similarity means that corresponding sides are proportional, leading to the equations a² = cy and b² = cx by using the cross product property. Finally, adding these, a² + b² = cy + cx, and applying the distributive property on the right side gives a² + b² = c(y + x), which simplifies to a² + b² = c² by substituting y + x with c, as they represent the same length.