Chapter 8.6, Problem 46PPS

a.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $PythagoreanTheorem$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:c^2={\left(a-x\right)}^2+h^2\end{array}$

Calculation:

Here, we will consider the following figure:

  

Now, in a right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Hence, the required answer is $PythagoreanTheorem$ .

b.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $Substituion\mathrm{Property}$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:c^2=a^2-2ax+x^2+h^2\end{array}$

Calculation:

Here, we will consider the following figure:

  

Now,

  $\begin{array}{l}{c}^2={\left(a-x\right)}^2+h^2\\ c^2=a^2-2ax+x^2+h^2\end{array}$

Hence, the required answer is $Substituion\mathrm{Property}$ .

c.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $PythagoreanTheorem$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:x^2+h^2=b^2\end{array}$

Calculation:

Here, we will consider the following figure:

  

Hence, the required answer is $PythagoreanTheorem$ .

d.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $Substitution\mathrm{Property}$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:c^2=a^2-2ax+b^2\end{array}$

Calculation:

Here, we will consider the following figure:

  

Now,

  $\begin{array}{l}Substituting{x}^2+h^2=b^{2}in{c}^2=a^2-2ax+x^2+h^2:\\ c^2=a^2-2ax+b^2\end{array}$

Hence, the required answer is $Substitution\mathrm{Property}$ .

e.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $Definitionof\cos ine$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:cosC=\frac{x}{b}\end{array}$

Calculation:

Here, we will consider the following figure:

  

Now, $\Delta ABC$ is a right triangle with acute $\angle A$ then the cosine of $\angle A$ is the ratio of the length of the leg adjacent $\angle A$ to the length of the hypotenuse.

Hence, the required answer is $Definitionof\cos ine$ .

f.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $Multiplication\mathrm{Property}$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:bcosC=x\end{array}$

Calculation:

Here, we will consider the following figure:

  

  $\begin{array}{l}Multiplyingeachsideof\cos C=\frac{x}{b}byb:\\ b\cos C=x\end{array}$

Hence, the required answer is $Multiplication\mathrm{Property}$ .

g.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $Substitution\mathrm{Property}$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:c^2=a^2-2a\left(b\cos C\right)+b^2\end{array}$

Calculation:

Here, we will consider the following figure:

  

  $\begin{array}{l}Substitutingb\cos C=xin{c}^2=a^2-2ax+b^2:\\ c^2=a^2-2a\left(b\cos C\right)+b^2\end{array}$

Hence, the required answer is $Substitution\mathrm{Property}$ .

h.

To determine

Verify each statement for the derivation of the law of cosines.

Answer to Problem 46PPS

  $Commutative\mathrm{Property}$

Explanation of Solution

Given information:

  $\begin{array}{l}hisanaltitudeof\Delta ABC\\ \mathrm{Prove}:c^2=a^2+b^2-2ab\cos C\\ \mathrm{Proof}:c^2=a^2+b^2-2ab\cos C\end{array}$

Calculation:

Here, we will consider the following figure:

  

  $\begin{array}{l}Commutativepropertyofaddition:\\ c^2=a^2-2a\left(b\cos C\right)+b^2\\ c^2=a^2+b^2-2ab\cos C\end{array}$

Hence, the required answer is $Commutative\mathrm{Property}$ .