Question

complete the steps below to prove the law of cosines. when filling in the blanks, you may use the letters a, b, c, x, and h. part 1: use the pythagorean theorem to find c². c² = b² - a² c²=(a - x)²+h² c²=(x + b)²+h² c²=(b - x)²+h² part 2: use the answer from part 1 to fill in the blanks. c²=x²+h²+□² - 2·□·□ part 3: use the pythagorean theorem to find a². a² = b² + h² a²=(c - x)²+h² a²=x²+h² a² = b² - x² part 4: use the answers from parts 2 and 3 to fill in the blanks. c²=□²+□² - 2·□·□ part 5: use trigonometry to fill in the blank. x = □cos c part 6: use the answers from parts 4 and 5 to fill in the blanks. c²=□²+□² - 2·□·□cos c

Explanation:

Step1: Apply Pythagorean theorem for $c^{2}$

In right - triangle $ADB$, by the Pythagorean theorem, $c^{2}=(b - x)^{2}+h^{2}$.

Step2: Expand $(b - x)^{2}$

Expand $(b - x)^{2}=b^{2}+x^{2}-2bx$. So $c^{2}=x^{2}+h^{2}+b^{2}-2bx$.

Step3: Apply Pythagorean theorem for $a^{2}$

In right - triangle $CDB$, by the Pythagorean theorem, $a^{2}=x^{2}+h^{2}$.

Step4: Substitute $a^{2}$ into the $c^{2}$ formula

Since $a^{2}=x^{2}+h^{2}$, then $c^{2}=a^{2}+b^{2}-2bx$.

Step5: Use trigonometry

In right - triangle $CDB$, $\cos C=\frac{x}{a}$, so $x = a\cos C$.

Step6: Substitute $x$ into the $c^{2}$ formula

Substitute $x = a\cos C$ into $c^{2}=a^{2}+b^{2}-2bx$, we get $c^{2}=a^{2}+b^{2}-2ab\cos C$.

Answer:

Part 1: $c^{2}=(b - x)^{2}+h^{2}$
Part 2: $c^{2}=x^{2}+h^{2}+b^{2}-2bx$
Part 3: $a^{2}=x^{2}+h^{2}$
Part 4: $c^{2}=a^{2}+b^{2}-2bx$
Part 5: $x = a\cos C$
Part 6: $c^{2}=a^{2}+b^{2}-2ab\cos C$