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 with missing justifications proves the pythagorean theorem using similar triangles: statement justification draw an altitude from point c to line segment ab let segment bc = a segment ca = b segment ab = c segment cd = h segment db = y segment ad = x y x = c c over a equals a over y and c over b equals b over x a2 = cy; b2 = cx a2 b2 = cy b2 a2 b2 = cy cx a2 b2 = c(y x) a2 b2 = c(c) a2 b2 = c2 which is not a justification for the proof? pieces of right triangles similarity theorem side-side-side similarity theorem substitution addition property of equality

Answer 1

The missing justification in the proof is substitution. The proof uses similar triangles and the Pythagorean theorem to show that a2 + b2 = c2.

Step-by-step explanation:

The missing justification in the proof is substitution. Here is the step-by-step explanation of the proof:

1. Draw an altitude from point C to line segment AB.
2. Segment BC = a, Segment CA = b, Segment AB = c (given information).
3. Segment CD = h, Segment DB = y, Segment AD = x (given information).
4. Using similar triangles, we can write the following ratios: c/a = a/y and c/b = b/x.
5. Cross-multiplying these ratios gives us a2 = cy and b2 = cx.
6. Substituting these equations into the equation a2 + b2 = c2, we get c2 = cy + cx.
7. Simplifying further, c2 = c(y + x).
8. Finally, factoring out c from the right side gives us c2 = c(c).
9. And simplifying one last time, we get c2 = c2.