Question

Consider the proof.

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

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) = 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?

a. Pythagorean theorem
b. definition of cosine
c. substitution
d. properties of multiplication

Answer 1

C. Substitution
substituting 2bx for 2bccos(a)

Answer 2

The Law of Cosines is used to find the remaining parts of a triangle when the triangle is not a right triangle and when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known.

The law of cosine states if a,b,c are the three sides of any triangle then
`a^(2) =b^(2) +c^(2) -2bccosA`

In the above proof in statement 8 the x in statement 5 is replaced by c cos A from statement 6.

Among the given options ,option C holds true as the reason for statement 8.