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 Pieces of Right Triangles Similarity Theorem 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) ? a2 + b2 = c(c) Substitution a2 + b2 = c2 Multiplication Which of the following is the missing justification in the proof?

A. Addition Property of Equality
B. Distributive Property of Equality
C. Transitive Property of Equality
D. Cross Product Property

Answer 1

The missing justification in the proof is Addition Property of Equality in the given right triangle.

Therefore, the correct answer is: option A). Addition Property of Equality

Step-by-step explanation:

To prove the Pythagorean Theorem, the proof constructs right triangles BDC and CDA using an altitude from point C to line segment AB.

In this case, the missing justification in the proof is Addition Property of Equality.

The 4 main properties of addition are commutative, associative, distributive, and additive identity. As per the addition property of equality, when we add the same number to both sides of an equation then the two sides remain equal. This can be written as, if a = b, then a + c = b + c.

It then labels the lengths of the segments of the right triangles. By proving that the ratios of corresponding sides in the similar triangles are equal, it shows that a2 = cy and b2 = cx.

Finally, using the Addition Property of Equality, it is concluded that a2 + b2 = cy + cx, which can be simplified to a2 + b2 = c2 using the Cross Product Property.