Geometric proportionality How can I calculate the height of the tree? I am with geometric proportionality.

 A: 
$FG \approx DE$ as $AG \approx AE$.
And $FG \approx AG$ as $DE \approx AE$ and as $BC \approx AC$.
Or perhaps most sophisticatedly: If $\frac {BC = 1.60}{CE=16}*k = DE=17.2$ then $\frac {DE=17.2}{EG=10} *k = FG$ and $\frac{BC=1.6}{CG = 16 + 10}*k = FG$.
A: I will denote by $a\cdot b$ the product of $a$ and $b$ (referred to as dot multiplication).

You want to find the gradient of the dotted line. Since it is straight, it is in linear form, namely, $$y=mx+b\quad\text{ or }\quad y=mx+c.\tag*{$\bigg(\begin{align}&\text{depending on how} \\ &\text{you were taught}\end{align}\bigg)$}$$

Firstly, you want to find the value $m$, since that is the gradient. Our unit of measurement is metres.
Now, according to the diagram: $$x_1=0\Rightarrow y_1=1.6\tag*{$\because$ $160$cm $=$ $1.6$ metres}$$ which is the height of the stick figure; and, $$x_2=16\Rightarrow y_2=17.2$$ which is the height of the smallest tree.

Secondly, we want to find $y$ at $x=16+10=26$, i.e. the height of the biggest tree.
To do this, we use the gradient formula: $$m=\frac{y_2-y_1}{x_2-x_1}$$ Now, we substitute the values of $x_{1,2}$ and $y_{1,2}$ in the following way: $$\begin{align} m&=\frac{17.2-1.6}{16-0} =\frac{15.6}{16}\\ &=0.975.\end{align}$$

Lastly, we find $b$ or $c$ (I will use $c$) by substituting $x=0$, since that is the $y$ intercept. It follows that, $$\begin{align}1.6&=0.975\cdot 0 + c \\ &= 0+c \\ \therefore c &= 1.6.\end{align}$$ $$\boxed{ \ \begin{align}\therefore y&=0.975\cdot 26 + 1.6 \\ &= 25.35+1.6\\ &=26.95.\end{align} \ }$$



$$\text{Ergo, the height of the biggest tree is $26.95$ metres.}\tag*{$\bigcirc$}$$


A: Welcome to Math.SE. It is always better if you show us the work you've done on a problem, and where you got stuck. I'll answer your question but please bear this in mind for next time. 
The length from the man to the base of the small tree is $16m$. The length from the man to the big tree is $26m(10m+16m)$
The ratio between these two is your geometric factor, $\frac{26}{16}=\frac{13}{8}=1.125$.
Simply multiply this factor by the height of the small tree to gain the height of the big tree. 
