Question

height of the triangle is h, and bd = 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=a² - x² c²=(a - x)²+h² c²=a²+h² c²=x²+h² part 2: use the pythagorean theorem to find b². b²=(x + a)²+h² b²=x²+h² b²=(a - x)²+h² b²=a² - c² part 3: use the answer from part 2 to fill in the blanks. b²=x²+h²+□² - 2·□·□ part 4: use the answers from parts 1 and 3 to fill in the blanks. b²=□²+□² - 2·□·□ part 5: use trigonometry to fill in the blank. x = □cosb part 6: use the answers from parts 4 and 5 to fill in the blanks. b²=□²+□² - 2·□·□cosb

Explanation:

Step1: Find $c^{2}$ using Pythagorean theorem

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

Step2: Find $b^{2}$ using Pythagorean theorem

In right - triangle $ADB$, $b^{2}=(a - x)^{2}+h^{2}$.

Step3: Expand $b^{2}=(a - x)^{2}+h^{2}$

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

Step4: Substitute $c^{2}=x^{2}+h^{2}$ into $b^{2}$ expression

We get $b^{2}=c^{2}+a^{2}-2ax$.

Step5: Use trigonometry

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

Step6: Substitute $x = a\cos B$ into $b^{2}$ expression

$b^{2}=a^{2}+c^{2}-2ac\cos B$.

Answer:

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