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 ? c over a equals a over y and c over b equals b over x pieces of right triangles similarity theorem a² = cy; b² = cx cross product property a² + b² = cy + cx 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?

1) Similarity Theorem
2) Cross Product Property
3) Addition Property of Equality
4) Substitution
5) Distributive Property of Equality
6) Multiplication

Answer 1

The missing step in the proof of the Pythagorean theorem involves the Substitution property. It justifies replacing y / x with c / c, as x and y together make up the hypotenuse c, completing the logical steps of the proof.

Step-by-step explanation:

The student's question concerns the proof of the Pythagorean theorem which relates the lengths of the sides in a right triangle. The missing step in the proof presented by the student involves translating the proportions obtained from the similarity of the smaller right triangles (produced by adding an altitude to the original triangle) back to the original sides of the triangle. Specifically, the justification for the statement 'y/x = c/c' is the Substitution property. This step recognizes that both x and y are parts of the hypotenuse c and when added together they equal c. Therefore, the substitution y + x for c is valid.

To elaborate, the triangles are all similar by the Similarity Theorem, which allows us to write the proportions c/a = a/y and c/b = b/x. The Cross Product Property is used next to rearrange these equations to a² = cy and b² = cx. The Addition Property of Equality allows us to add the two previous equalities together, resulting in a² + b² = cy + cx. Then, we substitute y + x with c, because the altitude splits the hypotenuse into these two segments. Finally, we use the Distributive and Multiplication properties to simplify and conclude that a² + b² = c².