Consider the proof. Given: In △ABC, BD ⊥ AC
Prove: the formula for the law of cosines, a2 = b2 + c2 – 2bccos(A)

Triangle A B C is shown. A perpendicular bisector is drawn from point B to point D on side A C. The length of B C is a, the length of D C is b minus x, the length of A D is x, the length of A B is c, and the length of B D is h.

Statement

Reason
1. In △ABC, BD ⊥ AC 1. given
2. In △ADB, c2 = x2 + h2 2. Pythagorean thm.
3. In △BDC, a2 = (b – x)2 + h2 3. Pythagorean thm.
4. a2 = b2 – 2bx + x2 + h2 4. prop. of multiplication
5. a2 = b2 – 2bx + c2 5. substitution
6. In △ADB, cos(A) = StartFraction x Over c EndFraction 6. def. cosine
7. ccos(A) = x 7. mult. prop. of equality
8. a2 = b2 – 2bccos(A) + c2 8. ?
9. a2 = b2 + c2 – 2bccos(A) 9. commutative property
What is the missing reason in Step 8?

Pythagorean theorem
definition of cosine
substitution
properties of multiplication