Question

geometric mean solve for all variables. 6. 7. 8.

Explanation:

Step1: Recall geometric - mean formula

In a right - triangle, if the altitude drawn to the hypotenuse divides the hypotenuse into two segments of lengths $a$ and $b$, and the length of the altitude is $h$, then $h^{2}=a\times b$.

Step2: Solve for $x$ in problem 6

We have $x^{2}=36\times16$. Then $x = \sqrt{36\times16}=\sqrt{36}\times\sqrt{16}=6\times4 = 24$.

Step3: Solve for $x$ in problem 7

We have $14^{2}=2\times x$. So $x=\frac{14^{2}}{2}=\frac{196}{2}=98$.

Step4: Solve for $x$ in problem 8

We have $x^{2}=27\times480$. First, $27\times480 = 12960$. Then $x=\sqrt{12960}=\sqrt{144\times90}=12\sqrt{90}=12\times3\sqrt{10}=36\sqrt{10}$.

Answer:

1. $x = 24$
2. $x = 98$
3. $x = 36\sqrt{10}$