Question

Need answer ASAP no small talk I have to add a second picture for the bottom

Answer 1

PythagoreanGiven the figure in the attached image.

1. Applying Pythagorean theorem to find c^2 ;

`c^2=(b-x)^2+h^2`

2. Using the answer above, let us simplify;

`\begin{gathered} c^2=(b-x)^2+h^2 \\ c^2=(b-x)(b-x)^{}+h^2 \\ c^2=b^2-2xb+x^2+h^2 \\ c^2=x^2+h^2+b^2-2xb \\ c^2=x^2+h^2+b^2-2\ast x\ast b \end{gathered}`

3. Using the Pythagorean theorem to find a^2 ;

`a^2=x^2+h^2`

4. combining the answers in 2 and 3 above;

`\begin{gathered} c^2=x^2+h^2+b^2-2\ast x\ast b \\ \text{and} \\ a^2=x^2+h^2 \\ \text{substituting;} \\ c^2=(x^2+h^2)+b^2-2\ast x\ast b \\ c^2=a^2+b^2-2\ast x\ast b \end{gathered}`

5. Applying trigonometry to solve for x;

`\cos \theta=(adjacent)/(hypotenuse)`
`\begin{gathered} \cos C=(x)/(a) \\ x=a\cos C \end{gathered}`

6. Using the answers from parts 4 and 5;

`\begin{gathered} c^2=a^2+b^2-2\ast x\ast b \\ \text{and} \\ x=a\cos C \\ \text{Substituting;} \\ c^2=a^2+b^2-2\ast x\ast b \\ c^2=a^2+b^2-2\ast(a\cos C)\ast b \\ c^2=a^2+b^2-2\ast a\ast b\cos C \end{gathered}`