Question

1. in $\triangle bdc, a^{2}=(b - x)^{2}+h^{2}$ 3. pythagorean thm.
2. $a^{2}=b^{2}-2bx+x^{2}+h^{2}$ 4. prop. of multiplication
3. $a^{2}=b^{2}-2bx+c^{2}$ 5. substitution
4. in $\triangle adb, \cos(a)=\frac{x}{c}$ 6. def. cosine
5. $c\cos(a)=x$ 7. mult. prop. of equality
6. $a^{2}=b^{2}-2bc\cos(a)+c^{2}$ 8. ?
7. $a^{2}=b^{2}+c^{2}-2bc\cos(a)$ 9. commutative property

what is the missing reason in step 8?
○ pythagorean theorem
○ definition of cosine
○ substitution
○ properties of multiplication

Explanation:

Step 7 establishes that $x = c\cos(A)$. Step 8 replaces $x$ in the equation from Step 5 ($a^2 = b^2 - 2bx + c^2$) with $c\cos(A)$, which is the substitution of an equivalent value into the equation.

Answer:

substitution