Find geometric mean and hence w in the following figure?

Mar 4, 2017

$w = 6$ and figure represents geometric mean between $12$ and $27$, which is $18$.

Explanation:

You have not stated the numbers or unknowns, whose geometric mean you seek to find.

However, image enclosed by you indicates that the geometric mean between $2 w$ and $4 w + 3$ is $3 w$, as

in such a figure geometric mean between $x$ and $y$ is $h$

or $\frac{x}{h} = \frac{h}{y}$ i.e. $x \times y = {h}^{2}$. Hence, in given problem

$2 w \times \left(4 w + 3\right) = {\left(3 w\right)}^{2}$

or $8 {w}^{2} + 6 w = 9 {w}^{2}$

or ${w}^{2} - 6 w = 0$

or $w \left(w - 6\right) = 0$ i.e. $w = 0$ or $w = 6$

As we are dealing with a geometric figure and $w$ cannot be zero (as this leads to $x = 0$ and $h = 0$), the answer is

$w = 6$ and figure represents geometric mean between $12$ and $27$, which is $18$.