Question

the height of the triangle is h, and cd = x. 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=(b - x)²+h² c²=b² - a² c²=b² - x² c²=x²+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 - c² a²=(b - x)²+h² a²=b² - x² a²=x²+h² 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 = □cosc part 6: use the answers from parts 4 and 5 to fill in the blanks. c²=□²+□² - 2·□·□cosc

Explanation:

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

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

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

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

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

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

Step4: Substitute $a^{2}$ into $c^{2}$ expression

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

Step5: Use trigonometry relation

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

Step6: Substitute $x$ into $c^{2}$ expression

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: $c^{2}=x^{2}+h^{2}+b^{2}-2\cdot b\cdot x$
Part 3: $a^{2}=x^{2}+h^{2}$
Part 4: $c^{2}=a^{2}+b^{2}-2\cdot a\cdot b$
Part 5: $x = a\cos C$
Part 6: $c^{2}=a^{2}+b^{2}-2\cdot a\cdot b\cos C$